Baitap Giaitich 1 Daihoc

BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 1 LI NI U. Trongnhiunm qua, cc cuc thi Olympic ton quc gia, quc t dnh cho hc sinh, sinh vin tr thnh mt sn chi tr tu nhm pht hin v m mmnhngtinngtonhctnglai.Quamtthisinhvinihcs phmtngnhiulnthamdcckthiOlympicton,bnthntihc tp cnhng iu tht qugivvn rnluyn t duy clp, sng to thngquavicgiiccbitonkh.Hnthna,xutphttnhiuamm vyuthchvilnhvcgiitchtonhc,tiluncmongmuntmti, tnghpnhngbitoncligiipvkhtrnnhngtpchtontrong ncvncngoi. Trn c snhng bi ton su tm c, tim rngn theonhiuhng khcnhau c nhng bi tonmilhn,hp dnhn. Nhm gip cc bn hc sinh , sinh vin ang n luyn chun b thi Olympic cthmmttiliuhtrchovicgiitoncamnh,tixinmnhdnvit cunsch:BitpgiitchdnhchoOlympicton.Mongrngquacun schny,ccbnstmthycnimvuivnhngcmxcringtrc nhngdngton,nhngbitonhaymlunaytrongnhnggiotrnhgii tch cn bn cc bn rt t gp. Nidungcunschnycchiaralm7chng.Tchng1n chng5,michngcchiaralm3phngm:Tmttlthuyt-Cc dngbitp(ckmtheoligiichitit)-Bitpngh.Chng6lh thngccbitptnghp-nngcaochoccchngtrnvinhngnh hng,gicchgii.Chng7lphngiithiuccthicaHiTon hc Vit Nam ra thi t nm 1993 n 2011. Vi kinh nghim cn non tr ca mt ging vin trong bui u dy hc, chc chn rng cun sch ny cn rt nhiu nhng sai st, rtmong s ch dy thmcaquthycgio,snggpcaccbnhcsinh-sinhvinyu thch ton ti rt ra c nhiu kinh nghim qu bu. Cui cng ti xin chn thnhcmnTh.SHunhTnTrnggingvinkhoaTon-Tin,trngi hc Qung Nam ng vin, ng h v gip cho ti trong vic hon thnh cun sch ny. Mi kin trao i xin bn c lin h theo a ch sau y: Vn Ph Quc, GV.Trng i hc Qung Nam, S 102- ng Hng Vng-TP. Tam KMail: [email protected] S in thoi: 0982 333 443 MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 2 CHNG 1 DY S THC V GII HN A. TM TTL THUYT 1. nh ngha dy sDy s l mt nh x: u N R ( ) n u n Ta thng k hiu dy l( )nuhoc { }nu . 2. Dy s hi t, phn k 2.1. nh ngha 2.1.1. nh ngha 1 a) Dy( )nuhi t naeR0 00, N , n > Nnu a c c > - e < N . K hiu:limnnu a= hoc( ) nnu a . b) Dy( )nukhng hi t th c gi l dy phn k. 2.1.2. Mnh 1 Gii hn ca mt dy hi t l duy nht. 2.1.3. nh ngha 2 a) Dy( )nuc gi l b chn trn nu:nnM u M - s eN. b) Dy( )nuc gi l b chn di nu:nm u m - > n eN. c) Dy( )nuc gi l b chn nu n va b chn trn va b chn di, tc l0:nu o o - > < n eN. 2.1.4. nh ngha 3 a) 0 0lim 0, N , n > Nn nnu A u A= + > - e > N . b) 0 0lim 0, N , n> Nn nnu B u B= < - e < N . Nhn xt: Tt c cc dy s c gii hn u phn k. 2.1.5. Mnh 2 a) Mi dy s tin n+ u b chn di. b) Mi dy s tin n u b chn trn. 2.2. Tnh cht v th t ca dy s hi t 2.2.1. Mnh 1 Cho( )nul mt dy s hi t c gii hn l a v hai s thc, o | . Nua o < N N . Nua | < N N . Nua o | < < < N N . 2.2.2. Mnh 2 MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 3 Cho( )nul mt dy s hi t. Khi : a) Nu 1 1: ,nN n n N u o - e e > > N Nthlimnnu o>b) Nu 2 2: ,nN n n N u | - e e > s N Nthlimnnu |s . c) Nu 0 0: ,nn n n n u o | - e e > s s N Nthlimnnu o |s s . 2.2.3. Mnh 3 Cho hai dy s( ) ( ) ,n nu vhi t Nu 0 0: ,n nn n n n u v - e e > s N Nthlim limn nn nu v s2.2.4. Mnh 4 Cho ba dy s( ) ( ) ( )n, , wn nu vsao cho: (i) 0 0 n, , wn nn n n n v u - e e > s s N N(ii) nlim limwnn nv a = = . Khi :limnnu a= . 2.2.5.Mnh 5 Cho hai dy s( ) ( ) ,n nu vsao cho: (i) 0 0 n, ,nn n n n u v - e e > s N N(ii)limnnu= +. Khi :limnnv= +. 2.2.6. Mnh 6 Cho hai dy s( ) ( ) ,n nu vsao cho: (i) 0 0 n, ,nn n n n u v - e e > > N N(ii)limnnu= . Khi :limnnv= . 2.3. Cc tnh cht v i s ca dy s hi t 2.3.1. Mnh 1 Cho hai dy s( ) ( ) ,n nu vv cc s, , a b eR. Khi , ta c: (i)lim limn nn xu a u a = = . (ii)( )limlimlimnnn nnnnu au v a bv b= + = += (iii)lim limn nn xu a u a = =MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 4 (iv) lim 0lim 00:nnn nnnuu vM v M= =- > s (v)( )limlimlimnnn nnnnu au v abv b= == (vi) 1 1lim 0 limnn nnv bv b = = = . (vii) limlimlim 0nnnnn nnu au av b v b= == =. 2.3.2. Mnh 2 Cho( ) ( ) ,n nu vl hai dy s thc. a)( )limlim: nnnn nnnuu vm v m= + + = +- > eN. c bit:(i)( )limlimlimnnn nnnnuu vv= + + = += + (ii)( )limlimlimnnn nnnnuu vv b= + + = += b) 0 0lim0, , ,nnnun n n n v e e= + - > - e e > >N N( ) limn nnu v= +. c bit: (i)( )limlimlimnnn nnnnuu vv= + = += + (ii)( )limlimlim 0nnn nnnnuu vv b= + = += > c) 1lim lim 0nn nnuu = + = . d) 0 0lim 01lim, , 0nnnnnuun n n n u= = +- e e > >N N. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 5 2.4. Cp s cng, cp s nhn 2.4.1. Cp s cng 2.4.1.1. nh ngha Cho dy s( )nuxc nh bi 1 01, n nu xu u d n+== + eN (0x , d l cc s hng s cho trc) c gi l cp s cng. Trong 0xgi l s hng u tin, d gi l cng sai. 2.4.1.2. Cc kt qu a) Cho( )nul cp s cng. Khi :( )11nu u n d = + n- eNb) Cho( )nul cp s cng. Khi : 1 22n n n nu u u+ += + eN . c) Cho( )nul cp s cng. Khi tng ca n s hng u tin l: ( )( )1112 12 2nnn kkn u uns u u n d=+= = = + ( . Ba s a, b, c theo th t lp thnh mt cp s cng2b a c = + . 2.4.2. Cp s nhn 2.4.2.1. nh ngha. Cho dy s( )nuxc nh bi: 1 01 nn nu xu u q+== eN (0x , d l cc hng s cho trc) c gi l cp s nhn. Trong 0xgi l s hng u tin, q gi l cng bi. 2.4.2.2. Cc kt qu a) Cho( )nul cp s nhn. Khi : 11nnnu u q -= eN . b) Cho( )nul cp s nhn. Khi : 21 2 n n nu u u+ += n eN. c) Cho( )nul cp s nhn. Khi tng ca n s hng u tin l: 111 q 11n nn kkqs u uq== = =. Ba s a, b, c khc khng theo th t lp thnh mt cp s nhn 20 b ac = > . 3. Tnh n iu 3.1. Dy n iu 3.1.1. nh ngha Cho( )nul mt dy thc. Ta ni rng: a)( )nutng 1nn nu u + s eN . b)( )nugim 1nn nu u+ s eN . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 6 c)( )nutng thc s1nn nu u + < eN. d)( )nugim thc s 1nn nu u+ < eN . e)( )nun iu ( )nu tng hoc gim. f)( )nun iu thc s( )nu tng thc s hoc gim thc s * Nhn xt (i)Nuccdy( )nu ,( )nv utng(tngnggim)th dy( )n nu v +tng ( tng ng gim). (ii)Nu cc dy( )nu ,( )nvu tng (tng ng gim) v cc s hng khng m th dy( )n nu vtng (tng ng gim). (iii)Mtdyscthkhngtnghockhnggim,Vddys ( )nuxc nh bi cng thc sau y:( ) 1nnu = , n-eN . 3.1.2. nh l a) Mi dy tng v b chn trn th hi t. b) Mi dy gim v b chn di th hi t. 3.1.3. Mnh a) Mi dy tng v khng b chn trn th tin n+. b) Mi dy gim v khng b chn di th tin n. * Nhn xt: (i)( )nutng limlimnnnnuu< + = +

. (ii) Nu( )nutng v hi t n a thsupnna ue=N. (iii) Nu( )nu tng th hin thin n b chn di bi 0u . 3.2. Dy k nhau 3.2.1. nh ngha Hai dy s( )nuv( )nvc gi l k nhau khi v ch khi: (i)( )nutng (ii)( )nvgim(iii)( ) lim 0n nnv u = . 3.2.2. Mnh 1 Nuhaidys( )nu v( )nv knhauthchnghitvccnggii hn.3.2.3. Mnh 2 ( Nguyn l Cantor) Cho hai dy s( ) ( ) ,n na bsao cho : (i) n na b sn eN (ii) | | | |1 1, ,n n n na b a b+ +c n eN (iii)( ) lim 0n nnb a =MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 7 Khi tn ti duy nhtaeRsao cho | | { },n nna b ae=N. Mt cch din t gn hn: Mi dy tht dn u c mt im chung duy nht. 4. Dy con 4.1. nh ngha Cho dy s( )nuv( )knl dy cc s t nhin tng thc s. Khi ta gi ( )knul mt dy con ca( )nu . 4.2. Mnh 1 lim limkn nn nu a u a = < + = . 4.3. Mnh 2 2 1 2lim lim limn n nn n nu a u u a+ = < + = = . 4.4. nh l Bolzano- Weierstrass. Mi dy s b chn u c th trch ra mt dy con hi t. 5. Dy Cauchy 5.1. nh ngha Dy( )nuc gi l dy Cauchy nu 0 00, n , ,n mm n n x x c c- > - e > < N . 5.2. Cc kt qu a)( )nul dy Cauchy*0 00, , pn n pn n n x x c c+ > - e > < e N N. b)( )nul dy Cauchy n hi t. 6. Dy chn, dy khng ng k, dy tng ng 6.1. Dy chn Dy( )nvchn dy( )nunu tn ti hng s C > 0 v tn ti s 0n eN sao cho 0n nn nu C v s > . Ta vit:( )n nu O b = . 6.2. Dy khng ng k Dy( )nukhng ng k so vi( )nvnu vi mi0 c >tn ti mt s nc eN sao chon nn nu vcc s > ,ngha l:lim 0nnnuv= . Ta vit:( )n nu o v =6.3. Dy tng ng Dy( )nu tng ng vi( )nvnu( )n n nu v o v = ,ngha llim 1nnnuv= . Ta vit n nu v ~ . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 8 7. Mt s loi dy quan trng 7.1. Dy truy hi truy hi cp 1 vi h s hng s a)Dng tng qut: 1 n ,a, bn nu au b+ = + e e N R. b) Cng thc + Nu1 a =th dy( )nul mt cp s cng. + Nu1 a =thAannu B = + .7.2. Dy truy hi tuyn tnh cp 2 vi h s hng s a) Dng tng qut: 2 1 nn n nu au bu+ += + eN ,, a beR. b) Cng thc: Xt phng trnh c trng ca dy: 20 a b = . + Nu phng trnh ny c hai nghim phn bit 1 2, thtn ti , A BeR sao cho: 1 2 nn nnu A B = + eN. + Nu phng trnh ny c nghim kpth tn ti, A BeR sao cho ( )nnu A Bn = + . + Nu phng trnh ny c nghim phcx iy = +th ta t 2 2r x y = = +,tan,,2 2yxt t | |= e |\ .. Khi ( ) os isin r c = + v( ) osn +Bsinnnnu r Ac =(,,n A Be e R N) . 7.3. Dy truy hi cp 1 dng:( )1,n nu f u n+ = * Cch lm + Bc 1: bin i a v dng: ( ) ( ) ( )( ) ( ) ( ),n nn nu f uu f u n

=

=. + Bc 2: t dy ph( )n nv u = . Khi ta thu c mt dy truy hi mi theo nvn gin hn. 7.4. Dy truy hi cp 2 dng :( )1 1, ,n n nu f u u n+ = * Cch lm + Bc 1: bin i a v dng: ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( )1 11 1,, ,n n n nn n n nu u f u uu u f u u n

+ =

+ = + Bc 2: t dy ph t dy ph( )n nv u = . Khi ta thu c mt dy truy hi mi theo nvn gin hn. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 9 8. Gii hn trn v gii hn di ca dy s 8.1. nh ngha a) Nu dy s( )nuc mt dy con ( )knusao cholimknnu a=th a c gi l mt gi tr ring ca dy( )nuv a c th hu hn hay l. b) Tp cc gii hn ring ca dy s b chn( )nuc gi tr ln nht.Gi tr ny c gi l gii hn trn ca dy( )nak hiu llimnnu. c) Tp cc gii hn ring ca dy s b chn( )nuc gi tr b nht.Gi tr ny c gi l gii hn di ca dy( )nak hiu llimnnu. 8.2. nh l 1 Mi dy s( )nuu c gii hn trn , gii hn di v { }{ }11lim limsup , ,... lim liminf , ,...n n nn nn n nnnu u uu u u+ +==. 8.3. nh l 2 Dy s( )nuc gii hn ( hu hn hay)lim limn nnnu u = . Khi :lim lim limn n nn nnu u u = = . 9.Gii thiu hai nh l quan trng v dy s 9.1. nh l Toeplitz Gi s ng thi xy ra cc iu kin sau y: (i)Cc s0 n,knkP-> eN . (ii) *11 nnnkkP== eN(iii)Vi mik-eNc nh,lim 0nknP+=(iv)limnnu a= < +. Khi dy( )nvxc nh bi ( )1 ,nn nk nkv P u n-== eN hi t v limnnv a= . 9.2. nh l Stolz Nu hai dy s( ) ( ) ,n nu vng thi tha mn cc iu kin sau: (i) *1 nn nv v+ > eNMATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 10 (ii)limnnv= + (iii) 11limn nnn nu uav v= th tn tilimnnnuav= . B- CC DNG BI TP 1.1. Cho dy s( )nuxc nh bi: 21arctan , n 12nun= > . Hy tnh tng 1 2 2011... S u u u = + + + . Gii Ta c: ( ) ( )( )( )2 22 1 2 11 2arctan arctan arctan2 4 1 2 1 2 1nn nun n n n+ = = =+ + =( ) ( ) arctan 2 1 arctan 2 1 n n + , 1 n > . Khi : 1 2 2011... S u u u = + + +arctan3 arctan1 arctan5 arctan3 ... arctan4023 arctan4021 = + + + =arctan4023 arctan1 arctan 40234t = . 1.2. Cho dy s( )nuxc nh bi : ( )21 !nu n n = + ,1 n >Hy tnh tng 1 2 2011... S u u u = + + + . Gii Ta c: ( ) ( ) ( ) ( )2 21 ! 1 ! 1 ! 1 !nu n n n n n n n n n n = + = + + = + ,1 n >Khi :1 2 2011... S u u u = + + +=1.2! 0.1! 2.3! 1.2! ... 2011.2012! 2010.2011! 2011.2012! + + + =1.3. Cho dy s( )nuxc nh bi : 211nun n=+ ,1 n > . Hy tnh tng 1 2 2011... S u u u = + + +. Gii BI TP RN LUYN K NNG TNH TON CC TNG HU HN MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 11 Ta c :21 11 1 1 112 .2 2 2 2nun n n nn n= =+ + + + + 21 1 1 12 2 1 11 12 22 2n nn nn n+ = = = + | |+ ++ |\ . ,1 n >Khi :1 2 2011... S u u u = + + +3 1 2011 20091 0 2 1 ... 1006 10052 2 2 2= + + + + + = 2011 1 2012 2011 11006 02 2 2+ + = . 1.4. Cho dy s( )nuxc nh bi : 221 1 11 2 1nnun n n+ | | | |= + + ||\ . \ .,1 n > . Hy tnh tng 1 2 20111 1 1... Su u u= + + + . Gii Ta c : 2 2 221 1 1 1 11 2 1 1 1 1 1nnun n n n n+ | | | | | | | |= + + = + + + + ||||\ . \ . \ . \ . Suy ra :2 22 22 21 11 1 1 11 11 11 11 1 1 11 1 1 1nn nun nn n| | | |+ + + ||\ . \ .= =| | | || | | |+ + + + + + || ||\ . \ .\ . \ . = ( ) ( )( ) ( )( )2 22 22 22 21 111 144n n n nn nn n n nn+ + + = + + + ( n 1 > ). MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 12 Khi :1 2 20111 1 1... Su u u= + + +

( )2 2 2 2 2 2 2 211 2 1 0 ... 2011 2012 2011 20104= + + + + + + += 2 22011 2012 14+ . 1.5. Cho dy s( )nuxc nh bi : 3 3 2 3 2 3 2 4 4 4 412 3 3 1nun n n n n n n n n=+ + + + + + + + + , 1 n >Hy tnh tng : 1 2 2011... S u u u = + + + . Gii Ta c : 3 3 2 3 2 3 2 4 4 4 412 3 3 1nun n n n n n n n n=+ + + + + + + + + = 4 4 4 411 1 1 1 n n n n n n n n + + + + + + + = ( ) ( )4 4 4 411 1 1 n n n n n n + + + + + + = ( )( ) ( )( )4 44 41 11 1 1 1n nn n n n n n n n+ =+ + + + ++ + = 4 41 n n = + ,1 n >Khi : 4 4 4 4 4 41 2 2011... 2 1 3 2 ... 2012 2011 S u u u = + + + = + + + = 42012 1 . -------------------------------------------------------------------------------------------- 1.6. Cho dy s( )nuxc nh bi: ( )( ) ( )( ) ( )( ) ( )0 11 23, 4

1 2 4 1 3 4 2 3, n 2 n n nu un n u n n u n n u = =+ + = + + + + > - Tnh 2011u ? BI TP XC NH CNG THC TNG QUT CA DY S MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 13 Gii Chia hai v ca (*) cho( )( )( ) 1 2 3 n n n + + +ta c: 1 24 43 2 1n n nu u un n n = + + +. t 3nnuvn=+ . Khi dy( )nvxc nh bi: 0 11 214 4 ,n 2n n nv vv v v = == > Phng trnh c trng c nghim l2 = . Do :( ) 2nnv A Bn = + . Vi 0 11 v v = = , ta c h: ( )1112 12AAA B B= = + = = ( ) ( )1 12 2 3 2 3 2n n n nn nv n u n n n = = + + . Vi n = 2011, ta c : 2011 201020112014.2 2011.2014.2 u = 1.7. Cho dy s( )nuxc nh bi : 0122011 2010, n 12010 2011nnnuuuu+=+= >+ Tnh 2011u? Gii Ta c : 111 1 40211 20102010 2011 1 1nnn n nuuu u u++ = = ++ . t 11nnvu=. Khi dy( )nvxc nh bi : 0114021 2010n nvv v+== + Khi :.4021nnv A B = + .Vi 01 v = , 16031 v =ta c h : 3124021 6031 12AA BA BB=+ = + == Do : 3 1 24021 12 2 3.4021 1nn n nv u = = + Vi2011 n = , ta c : 20112011 2011 20112 3.4021 113.4021 1 3.4021 1u+= + = . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 14 1.8. Cho dy s( )nuxc nh bi : 012532011 2012 , n 1nn nuu u+== + >. Tnh 2011? uGii t2012nn nv u = . Khi dy( )nvxc dnh bi : ( )0 0011 112522011, n 1 2012 2011 2012 2012n n nn n n nv uvv v v v++ += = = > + = + + Suy ra: ( )21 2 02011 2011 ... 2011 252. 2011n nn n nv v v v = = = = = . Do : ( )252. 2011 2012n nnu = + . Vi2011 n = , ta c: ( )2011 20112011252. 2011 2012 u = + . 1.9. Cho dy s( )nuxc nh bi : 121122 2 1, n 22nnuuu= = >. Tnh 2011u? Gii Ta c : 11sin2 6ut= = , 222 2 1 sin6sin2 2.6utt = = Chng minh bng quy np ta c : 1sin2 .6n nut=Vi2011 n = , ta c : 2011 2011sin3.2ut= . 1.10. Cho dy s( )nu( n = 1, 2, ...) c xc nh bi :

121121 1, n 22 4n n n nuu u u+=| |= + + > |\ . Tnh 2011? u MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 15 Gii Ta c : 1 1 1 11 1 1ot cot2 2 4 2 2u ct t+= = = 221 1 1 1 1 1 1 1ot cot ot2 2 4 4 4 4 2 2 4 2sin4u c ct t tt| || ||= + + = + | |\ . |\ . 22 2 12cos os 1 os1 1 1 1 18 4 4cot4 4 4 2 2sin sin sin 2sin os4 4 4 8 8c cct t ttt t t t t+| |+ |= + = = = | |\ . Chng minh bng quy np ta c : 11cot2 2n n nut+= . Vi2011 n = , ta c : 2011 2011 20121cot2 2ut= . 1.11. Cho dy s( )nuxc nh bi: 121112, n 22nnnuuuu=+ = >. Hy xc nh cng thc tng qut ca dy s( )nu . Gii Xt hai dy s( )nxv( )nyxc nh nh sau : ( )( )1 12 21 11 12, 12 2 2 2n n nn n nx yx x y ny x y n = == + >= > Chng minh bng quy np : nnnxuy=1 n > . Vn l by gi chng ta i tm cng thc tng qut ca hai dy( ) ( ) ,n nx yl xong. rng:( )( )22 22 21 11 11 121 11 11 12 22222 2 22 2n n n nn n nn n nn n nn n nn n n nx y x yx x yx x yy x yy x yx y x y + = + = + = + == = MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 16 ( )( )2222 222 22 22 2n n n nn n n nx y x yx y x y + = + = ... ( ) ( )( ) ( )1 11 12 21 12 21 12 2 2 22 2 2 2n nn nn nn nx y x yx y x y + = + = + = = .y l mt h phng trnh theo hai n,n nx y .Gii h trn ta c: ( ) ( )( ) ( )1 11 12 22 212 2 2 2212 2 2 22 2n nn nnnxy (= + + ( (= + ( . Vy ta thu c : ( ) ( )( ) ( )1 11 12 22 22 2 2 222 2 2 2n nn nnx + + =+ -------------------------------------------------------------------------------------------- 1.12. Cho cc s thc dng 1 2 2011, ,..., x x xtha mn iu kin: 201112011kkx=>. t 20111nn kku x==. Chng minh rng dy( )nutng. GiiVi x > 0 ta lun c: ( )( ) 1 1 0nx x > . iu ny tng ng vi 11n nx x x+ > . Do : 2011 2011 2011111 1 12011 0n nn n k k kk k ku u x x x++= = = = > > . Vy dy( )nutng. 1.13. Cho dy s( )nuc xc nh nh sau: 20111 1 2010120120 , 2011u 2010n nnu uu> =2, 3,... n =Chng minh rng dy s( )nugim v b chn di bi2012. BI TP V CHNG MINH TNH N IU, B CHN CA DY S MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 17 Gii T20111 201012012 2011u 2010n nnuu = , 2, 3,... n =Suy ra: 20111 201011 201220102011n nnu uu| |= + |\ . R rng0 nu-> eN . p dng bt ng thc Cauchy cho 2011 s dng ta c: 2011 20111 1 1 2011 20101 120101 2012 1 20122010 ...2011 2011n n n nn nu u u uu u | || | | = + = + + + | |\ .\ ._2012 > . Li c:( )201120111 11 2012 12010 2010 1 12011 2011nn nuu u | |= + s + = |\ .(do2012nu > ). Vy( )nugim v b chn di bi2012. 1.14. Cho dy s( )nuxc nh bi: ( ) ( )01 102011 1 , t 0,1,nn n ttnutu t uu-+ >= + e eN Chng minh dy( )nuhi t Gii Xt hm s:( ) ( )120111tttf x t xx= +,( ) ( ) 0,, t 0,1 xe + e . Ta c:( ) ( )11 1 2011tf x t x | |' = |\ .. ( ) 0 2011tf x x ' = = . Lp bng bin thin, ta d dng suy ra:( ) 2011tf x > . M( )1 n 2011 ntn n nu f u u= e > e N N hay 12011tnu >Do : 1 n nu u+ = ( )1 1120111 2011 0tt tn n n n ttntt u u tu uu| | + = s |\ . ,( ) 0,1 t eDy( )nugim v b chn di bi2011t nn hi t. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 18 1.15. Cho dy s( )nuxc nh bi: 2.4nn n nnu C =,1 n > . Chng minh dy s( )nuhi t. Gii Lp t s: ( )( ) ( )( )( )( )122 2 112 121.2 2 ! !1 4 2 14. . .4 2 !2 1 1 !.4nnn nnnn nn nnCn nu n nu n n nn n nC++ ++++++ += = =+ + = 211 1 n4 4 n n-+ > e+N . Vy dy( )nutng thc s. Hn na: 12 2 21 1 1 1 1 1 1ln ln 1 ln 14 4 2 4 4 8 8 8 1kkuu k k k k k k k k+| | | |= + = + < = ||+ + + +\ . \ . ( )1 1 1 1111 1 1 11 1 1 1 1 1ln ln ln8 1 8 1n n n nkk kk k k kkuu uu k k k k ++= = = =| | | | < < ||+ +\ . \ . Hay 811 1 1 1 1 1 1ln ln 1 ln ln 1 ln8 2 8 2 8 2n n neu u u un n| | | | < < + < + < ||\ . \ .. Vy( )nul dy hi t. 1.16. Cho 1 2 2011, ,..., x x xl cc s thc dng c nh. Xt dy s : 1 2 2011...2011n n nnnx x xu+ + += ,n-eNChng minh rng dy( )nutng . Gii t 1 2 2011...2011n n nnx x xv+ + += . p dng bt ng thc Bunhiacovski ta c : 21 1 1 1 1 122 2 2 2 2 21 1 2 2 2011 2011 21 1 1 1 1 11 2 2011 1 2 20111 11...2010... ....2010 2010n n n n n nnn n n n n nn nv x x x x x xx x x x x xv v+ + + + + + + | |= + + + |\ .+ + + + + +s = MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 19 Ta s chng minh( )nul dy tng Li p dng bt ng thc Bunhiacovski ta c : ( )2 2 222 21 2 20111 1 2 2011 2 1 2 21 ...1. 1. ... 1.2011 2011x x xu x x x u u u+ + += + + + s = s . Gi s rng 1 n nu u s . Khi : 111 1nn n nn n n nv v v v s sTa c : 1 2 21111 1 1 11nnn n n n nnn n n n n n nn nnv vu v v v uvv+++++ + += > > = = =Vy( )nul dy tng. 1.17. Cho dy s( )nuxc nh bi: 1101, n 1n nuu u == + > Chng minh rng: 1 2 20112011...2u u u + + + > . Ta c: 11, k =1, 2,3,...,nk ku u+ = +Suy ra : 2 2 2 21 11 1 12 1 2n n nk k k k k kk k ku u u u u u n+ += = == + + = + + 211 10 22n nn k kk knu u n u+= =s = + > . Cho n = 2011 , suy ra : 1 2 20112011...2u u u + + + > . 1.18.Cho dy s( )nuxc nh nh sau :( )( )2 1 112nn n nu+ + +=,1 n >Chng minh rng : 1 2 20112011...2013u u u + + + < . Gii Ta c : ( )( )( )2 122 12 1 1kk kukk k k+ = =++ + +. p dng bt ng thc Cauchy ta c :( ) ( ) 2 1 1 2 1 k k k k k + = + + > + . BI TP V DY S V BT NG THC MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 20 Suy ra : ( )1 1 11 1kk kuk k k k+< = + +. Do : 211 21 12 14 4kiikuk kk k =< < =+ ++ +. Cho2011 k =ta c : 1 2 20112011...2013u u u + + + < . 1.19. Cho( )nul dy s thc dng tha : 21 , n 1n n nu u u +s > . Chng minh rng: 1, n 1nun< > . Gii +Vi1 n = , 21 1 2 1 1111u u u u u s < < = ( ng trong trng hp ny) +Vi2 n = , 222 1 1 11 1 1 14 2 4 2u u u u| |s = s < |\ . ( ng trong trng hp ny) + Gi s khng nh trn ng n n. Ta s chng minh n ng n1 n + . Tht vy! Xt hm s:( )2f x x x = . R rng( ) f xl hm s tng trn 10;2 ( ( . Do ( )( )1 2 21 1 1 1 1 11 1 1n nu f u fn n n n n n n+| |s s = = < |+ + +\ .. Vy 1, n 1nun< > . 1.20. Cho dy( )nuxc nh bi: 021120122012n nnuu uu += =. Chng minh rng: 20122011 14023 2x < < . Gii R rng:0 1nnu < < eN. Ta c: 211 1 1 11 1 1 1 12012,2012 2012 2012 2011nn n nnn n n n n n n nuu u uvu u u u u u u u++ + + + | |= = = = = e |\ . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 21 Suy ra: 2012 12012 01 1 2012 2012 4023... 1 ,1 2,2012 2011 2011v vu u| | | |= + + + e + + = ||\ . \ . Vy 20112011 14023 2u < Chng minh rng: 1 2... 2nu u u + + + < . Gii Vik-eN , ta c: ( )231 1 1 11 1kk kk k k k k kk k| |= = = |+ + +\ . + 1 1 1 11 1kk k k k| || |= + | |+ +\ .\ .= 1 1 1 11 21 1 1kk k k k k| || | | |= + < | ||+ + +\ . \ .\ . Do : 1 21 1 1 1 1 1.. 2 1 ... 2 1 22 2 3 1 1nu u un n n| | | |+ + + < + + + = < ||+ +\ . \ . 1.22. Cho dy s( )nu ,( )nvxc nh nh sau: 1 1110 , v 01, n 11, n 1n nnn nnuu uvv vu++> >= + >= + > Chng minh rng:( )2332011 20112 2011 u v + > . Gii t( )2wn n nu v = + . Khi :( ) ( )221 1 11 1wn n n n nn nu v u vu v+ + + (| |= + == + + + (|\ . = MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 22 =( ) ( )22 1 1 1 12n n n nn n n nu v u vu v u v| | | |+ + + + + + ||\ . \ . >( ) ( )2 1 12 w 8n n n n nn nu v u vu v| |+ + + + > + |\ .( bt ng thc Cauchy) Suy ra:( )1 2 2w 8 w 2.8 ... w 2 .8n n nw n > + > + > > + M( )222 1 11 11 1w 2 2 16 u vu v ( | | | |= + + + > + = (||\ . \ . ( bt ng thc Cauchy) V th:wn > ( ) 16 2 .8 8 n n + =33w 2nn >hay( )2332n nu v n + >Chn2011 n = , ta c:( )2332011 20112 2011 u v + > . 1.23. Cho a thc( )4 3 22 3 2 2 f x x x x x = + + + +v dy s( )nuxc nh bi:( )( )12 12nnkf kuf k==[. Chng minh: 1 2 20112011...4024u u u + + + = < |\ .. Mt khc1nu > < = |+ + + +\ .

1 1 112nn nun n n+ > > = +. Vy 11 1nun < < . 1.25. Cho dy s( )nuxc nh nh sau: 12112 , n 1n n nuu u u+== + >. Chng minh rng: 1 2 20111 1 11 ... 21 1 1 u u u< + + + tng. Mt khc: 2 33 21 , u 14 16u = = >suy ra: 201211n 3 1nuu> > < . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 24 Ta c phn tch sau:( )11 11 1 1 1 1 111 1n n nn n n n n nu u uu u u u u u++ += + = = + + ( )2010 20101 11 1 2012 20121 1 1 1 1 12 1,21n nn n nu u u u u u= =+| | = = = e |+\ . . Vy 1 2 20111 1 11 ... 21 1 1 u u u< + + + - e s < + N . n eN, chia n cho m ta c:n qm r = +, *,0 1 q r m e s s N . Khng mt tnh tng qut c th xem 00 u = . Ta c: .1 2n m r m r rn qm r mu qu u u qm u uu u qu r k kn qm r m qm n nc++= s + s s = + < + ++ +V 0 1 r m s s nn rub chn. Do 0N - eN sao cho 0n N >th: 02runcs < . T suy ra: 00n > Nnuknc s < + . Vy *lim inf ,n nnu unn n = e ` )N . 1.31. Tm 31lim sin!n nnkn kn n n=[. Gii + Bng cch s dng tnh cht n iu ca hm s, ta d dng chng minh c bt ng thc: 3 3 5sinx < x6 6 5!x x xx < +,0 x >Vn dng kt qu vo bi ton ny, ta c nh gi nh sau: 2 3 2 43 3 61 1 11 sin 16 ! 6 5!n n n nk k kk n k k kn n n n n n= = =| | | | < < + ||\ . \ .[ [ [ + Trc ht, ta tnh 231lim 16nnkkn=| | |\ .[ ? Cng s dng tnh cht n iu ta chng minh c bt ng thc: ( ) ln 11 , x 01xx x xx< + < > =+. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 27 V th, ta c nh gi: 2 2 23 2 3 31 1 1ln 16 6 6n n nk k kk k kn k n n= = =| |< < | + \ . [ Hay ( )( )( )( )( )23 3 3 211 2 1 1 2 1ln 16 36 6 6nkn n n n n n kn n n n=+ + + + | |< < | +\ .[. p dng nh l kp ta suy ra c 1 2 2183 31 11lim ln 1 lim 16 18 6n nn nk kk ken n = =| | | | | | = = | ||\ . \ .\ .[ [. + Tip theo ta s chng minh 1 2 4183 61lim 16 5!nnkk ken n=| | + s |\ .[. Tht vy!2 4 2 43 6 3 61 1ln 16 5! 6 5!n nk kk k k kn n n n= =| | | | + < + = ||\ . \ . [ =( )( )( )( )( )23 61 2 1 3 3 11 2 136 30.5!n n n n nn n nn n+ + + + + +Suy ra: 1 2 4183 61lim 16 5!nnkk ken n=| | + s |\ .[ Vy cng theo nh l kp ta suy ra: 1 3181lim sin .!n nnkn ken n n==[ 1.32. Cho( )nul dy cc nghim lin tip ca phng trnh lng gic t anx x =, x > 0. Tm 1lim os4n nnu uc+ | | |\ . ? Gii D thy; n = 1, 2,... lim2n nnn u n utt t< < + = +. Hn na: 1limtan lim 02 tannn nnn uutt | |+ = = |\ . lim 02nnn utt| | + = |\ . ( dot anx y =l hm lin tc) Do :( ) ( )1 1lim lim 1 02 2n n n nn nu u n u n ut tt t t+ + ( | | | | = + + + = || (\ . \ . Vy t tnh lin tc ca hm scos y x = , ta suy ra: MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 28 12lim os os .4 4 2n nnu uc ct+ | | = = |\ . 1.33. Cho dy s thc dng( )nuc xy dng nh sau: 11201110 , n 2nn kkuu u=> = > Chng minh rng dy nun| | |\ . c gii hn hu hn khin v hy tm gii hn . Gii Ta c: 2011 2011 201111 nn k n n nku u u u u+== = + >( do *0 nnu > eN ). Suy ra: 1 2n nu u n+ > >hay dy( )nutng thc s. Mt khc: ( ) ( )2011 2011 2010 2010 2010 20101 1 11 1 1 n 2n n n n n n n n nu u u u u u u u u+ + += + = + < + < + >Chng minh bng quy np n gin ta c: 2010 20101 21nu u n+< + .Suy ra:( )nub chn trn. Do ( )nuc gii hn hu hn khin . Hn na ( )( )2010 20101 2201010 0 01 11nu u nnn nn+| | | |< < + ||+ +\ . \ . +. Vylim 0nnun= . 1.34.Cho dy s( )nuxc nh bi: ( ) ( )( )1213 23 2 2 6 5 27 18, n 1 *n n nuu u u+= += + + > t 11, 2nnkkv nu-== e+N . Hy tmlimnnv. Gii Nhn c hai v ca (*) cho3 2 +ta c: ( ) ( )213 2 3 2n n nu u u++ = + 3 MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 29 ( )( )( )2213 2 2 3 3 3 0 nn n n n nu u u u u-+ + = + = > eN( )*1nn n nu u u+ > e Nl dy s tng. Hn na 13 2nu u > = + . Gi s( )nub chn trn. Khi dy( )nuhi t. t ( )lim3 2nnL u L= > + . Chuyn gi thuyt bi ton qua gii hn ta c: ( ) ( ) ( )223 2 2 6 5 27 18 3 0 3 L L L L L = + + = =( iu ny v l). Vylimnnu= +. Ta li c:( )( ) ( )( )( )( )113 2 3 3 21 3 2 1 13 3 23 2k k kk k kk ku u uu u uu u+++ = ++ = = + + 11 1 12 3 3k k ku u u + = + 1 111 1 12 3 3n nnk kk k kvu u u= =+| | = = |+ \ . 1 1 11 1 2 12 3 3 3n nu u u+ += = 2lim2nnv = . 1.35. Cho dy s( )nuxc nh bi: ( )12 2 21 2011ln 2011 2011, n 12010n nu au u+= = + >. Chng minh rng dy s( )nuc gii hn khin + v tm gii hn . Gii Xt hm s:( ) ( )2 2 22011ln 2011 20112010f x x = + MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 30 ( )( )2 22011 2 2011 2 1. .2010 2011 2010 2.2011 20101, 2010x x xf xx x xf x x' = s =+' s eR t( ) ( ) g x x f x = ( ) ( ) 1 0g x f x x ' ' = > eR. Suy ra:( ) y g x =tng thc s trnR v y l hm lin tc. V ( ) ( )22011 0 0 g g 00 x x o < th 0xl im cc tiu ; n chn v ( )( )00nf x = > . Chng minh rng tn ti( ) c a;b esao cho( ) f c 0 =v( ) f c 0 ' s . Gii T gi thit suy ra f bng 0 ti t nht mt im trong khong( ) a; b . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 70 t( ) () { }c inf x a;b : f x 0 = e = , ta c( ) f c 0 = . V( ) f a 0 ' >nn() ( ) f x 0x a;c > e . Hn na( ) f c 'tn ti nn ( )( ) ( ) ( )h 0 h 0f c h f c f c hf c lim lim 0h h + +' = = s . 3.8. Gi s f c o hm trn mt khong cha| | 0,1 ,( ) ( ) 0 0 , f 1 0 f ' ' > < . Chng minh rng tn ti( ) ( ) ( ) | |0 00;1 : x 0;1 x f x f x e s e . Gii f c o hm trn mt khong cha | | 0,1| | ( ) ( )| |( )0 00,10;1 : maxxx f x f x f xe - e s = . Ta s chng minh: 0 00,x 1 x = = . Tht vy! ( ) ( )( ) ( )( ) ( )( |00 0lim 0 0 0;1 : 0x 0;xf x f f x ff h hx x+ ' = > - e > e( ) ( ) ( | ( ) 0 x 0; 0 f x f h f > e khng phi l gi tr ln nht ca( ) f xtrn| |00,1 0 x = . ( ) ( )( ) ( )( ) ( )| )111 1lim 1 0 0;1 : 0x ;11 1xf x f f x ff k kx x ' = < - e < e ( ) ( ) | ) ( ) 1 x ;1 1 f x f k f < e khng phi l gi tr ln nht ca( ) f xtrn | )0;1 1 k x = . 3.9. Cho mt hm s f xc nh trnR tho mn ( ) ( ) 0 0 , f sin x f x x = > eR. Chng minh rng o hm ca f ti 0 khng tn ti. Gii Gi s( ) 0 f 'tn ti.0;2xt| | e |\ . ta c: ( ) ( )( )( ) ( )0 00 0 sin sin0 lim lim 10 0x xf x f f x f x xfx x x x+ ++ ' > = > = . Tng t ta cng chng minh c ( )10 1 f' < iu ny chng t( ) 0 f 'khng tn ti. ------------------------------------------------------------------------------------------------ BI TP VO HM CP CAO MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 71 3.10. Chng minh rng:() f x arctan x =tho mn phng trnh: ( )()() ( )( )() ( )( )( )()n n 1 n 2 21 x f x 2 n 1 xf x n 2 n 1 f x 0 + + + =vix eR v n 2 > .Gii () f x arctan x =() ( ) ()221f x 1 x f x 11 x' ' = + =+(1) Ly o hm hai v ca (1) suy ra: ( ) () ()21 x f x 2xf x 0 '' ' + + = . Bng quy np ta chng minh c: ( )( )() ( )( )() ( )( )( )()n n 1 n 2 21 x f x 2 n 1 xf x n 2 n 1 f x 0 + + + = ( ) x ,n 2 e > R+ Mnh ng trong trng hp n = 2. + Gi s mnh ng nn k =tc l: ( )( )() ( )( )() ( )( )( )()k k 1 k 2 21 x f x 2 k 1 xf x k 2 k 1 f x 0 + + + =(*) Ly o hm hmhaiv ca (*) ta c ()() ( )( )() ( )( )()( )( )() ( )( )( )()( )( )()()() ( )( )()k k 1 k 1 2k k 1k 1 k k 1 22xf x 1 x f x 2 k 1 f x2 k 1 xf x k 2 k 1 f x 01 x f x 2kxf x k 1 kf x 0+ + + + + + + = + + + =

3.11. Cho f l hm kh vi n cp n trn( ) 0;+ . Chng minh rng vix 0 > , ()( )() nnn n 1n 11 1 1f 1 x fx x x+| | | | | |= |||\ . \ . \ . Gii + Mnh ng trong trng hpn 1 = . + Gi s mnh ng trong trng hpn k s , tc l:

()( )() kkk k 1k 11 1 1f 1 x fx x x+| | | | | |= |||\ . \ . \ . + Ta s chng minh mnh trn ng vin k 1 = + . Tht vy! ( )( )( )()( )()kk 1 kk 1 k k 1k k k 1 k 21 1 1 11 x f 1 x f 1 kx f x fx x x x++ + | | '| | | | | | | | | | | | | | |' = = ||||||| |\ . \ . \ . \ . \ . \ . \ .\ . =( )( )( )() k kk 1 k 1k 1 k 21 11 k x f 1 x fx x+ + | | | | | | | |' ||||\ . \ . \ . \ . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 72 ( )( )( ) kk 1k k 2k 1k 1 1f 1 x fx x x+| | | | | |' = |||\ . \ . \ .. Li c:( )( )( )( ) k k 1k 1 k 1k 2 k 21 11 x f 1 x fx x '| || | | | | | | |' ' = | |||| |\ . \ . \ . \ .\ . Theo gi thit quy np vi trng hpn k 1 = ta c: ( )( )( ) k 1k 1k k 2k1 1 1f 1 x fx x x| | | | | |' = |||\ . \ . \ ..T suy ra( )( )( )k 1k 1k 1 kk 21 1 11 x f fx x x++++| | | | | | = |||\ . \ . \ .. Vy bi ton c chng minh xong 3.12. Cho fkh vi trn( ) a; bsao cho vi( ) x a;b eta c:() () ( )f x g f x ' = , trong g ( ) C a;be . Chng minh fCe ( ) a; b . Gii Ta c:() () ( ) () () ( ) () () ( ) () ( )f x g f x f x g f x f x g f x g f x ' '' ' ' ' = = =() () ( ) () ( ) ( ) () ( ) ( ) () ( )2 2f x g f x g f x g f x g f x ''' ' ' = +Do f, f '' '''u lin tc trn( ) a; b . Chng minh bng quy np ta c ()( )nf n 3 >u l tng cc o hm ()( )kg fvik 0; n 1 = . T suy ra iu phi chng minh. ------------------------------------------------------------------------------------------------ 3.14. Cho| | f : ; 1;12 2t t ( ( l mt hm kh vi c o hm lin tc v khng m. Chng minh tn ti 0x ;2 2t t| |e |\ . sao cho( ) ( ) ( ) ( )2 20 0f x f x 1 ' + s . Gii Xt hm s: ()g : ; ;2 2 2 2x arctanf xt t t t (( (( BI TP VNH L GI TR TRUNG BNH MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 73 g l hm lin tc trn;2 2t t ( ( . Nu() f x 1 = th g kh vi ti mi x v ()()()2f xg x1 f x'' =. Nu tn ti 0x ;2 2t t| |e |\ . sao cho ( )( )00f x 1f x 1=

=

th 0xl cc tr a phng ca hm f nn theo nh l Fermat ta suy ra c( )0f x 0 ' = . V th ta c: ( ) ( ) ( ) ( )2 20 0f x f x 1 ' + = . Nu() f x 1x ;2 2t t| |= e |\ . th p dng nh l Lagrange cho hm g trn on;2 2t t ( ( ta c :( )( ) ( )0020f xx ; : g g2 2 2 2 2 21 f xt t t t t t'| | | | | | | | | |- e = |||||\ . \ . \ . \ .\ .. D thy: ( )( ) ( )020f x01 f xt t's s. Vy ta chng minh c( ) ( ) ( ) ( )2 20 0f x f x 1 ' + s . 3.15. Cho f l mt hm thc kh vi n cp1 n +trnR. Chng minh rng vi mi s thc,, a < b a btho mn ( ) ( )( )( )( ) ( )( )()...ln...nnf b f b f bb af a f a f a| | ' + + += |' + + +\ . tn ti( ) ; c a b esao cho ( )( ) ( )1 nf c f c+= .Gii Vi a, b l s thc,a b . Chng minh phng trnh( )21' = f xx c nghim ln hn 1 Gii t( ) ( )1g x f xx= fkh vi lin tc trn( ) ( ) ( )11; lim 1 0xf x f++ = =( ) ( )1 11lim lim 0x xg x f xx+ + | | = = |\ .. ( ) ( ) ( ) ( )1 10 lim 0 lim lim 0x x xf x f x g x f xx x+ + +| |s s = = = |\ . ( ) ( ) ( ) ( )0 01lim lim 1; : 0x xg x g x x g x++ ' = - e + =hay( )0 201f xx' = . Vy phng trnh( )21' = f xx c nghim ln hn 1. BI TP VMT S NGDNGCA O HM MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 76 3.20. Chng minh phng trnh: 2010sin3x 2011cos 2x 2012cos x sin x 0 + + + =c nghim trnR. Gii Xt hm s:()2010 2011f x cos3x sin2x 2012sin x cos x3 2= + + Hm sflin tc trn| | 0;2t , kh vi trn( ) 0;2t . Theo nh l Lagrange tn ti( ) c 0;2t esao cho: ( )( ) ( ) ( ) f 2 f 0 2011 2012 2011 2012f c 02 0 2tt t + +' = = = 3.21. Gi s: f R R c o hm cp 2 tho mn:( ) ( ) 0 1,f 0 0 f ' = =v ( ) ( ) ( ) | ) 5 6 00; f x f x f x x '' + > e + . Chng minh rng: ( )2 33 2x xf x e e > ,| ) 0; x e + . Gii Ta c: ( ) ( ) ( ) | ) 5 6 00; f x f x f x x '' ' + > e +( ) ( ) ( ) ( ) ( ) | ) 2 3 2 00; f x f x f x f x x '' ' ' > e +t( ) ( ) ( ) | ) 2, x 0; g x f x f x ' = e + . Khi ( ) ( ) | ) ( ) ( ) | )33 0 , x 0; 0 ,x 0;xg x g x e g x '' > e + > e + ( )3xe g xtng trn| ) 0; + () ()3 22 2x x xe g x e g x e > > () ( ) | ) ( ) ( ) | )2 22, x 0; 2 0 x 0;x x x xe f x e e f x e ' ' > e + + > e + ( )22x xe f x e+tng trn| ) 0; +( ) ( ) | )2 0 02 0 2 3 ,0;x xe f x e e f e + > + = +( )2 33 2x xf x e e > , | ) 0; x e + . 3.22 Cho: f R R l hm kh vi cp hai vi o hm cp 2 dng. Chng minh rng:( ) ( ) ( ) f x f x f x ' + > vi mi s thc x. Gii + Nu( ) 0 f x ' =th( ) ( ) ( ) f x f x f x ' + = vi mi x : hin nhin. + Nu( ) 0 f x ' l hm tng( ) ( ) 0 f c f x ' ' < < . V vy ( ) ( ) ( )0 f x f x f x ' + < . + Nu( ) 0 f x ' >th chng minh tng t nh trng hp( ) 0 f x ' = . Vy( ) 1 cos cos 11x xx xt t+ >+| ) 2; x e + . 3.24. Gi s( ) f xkh vi trn( ) ; a bsao cho( ) lim,limx a x bf x+ = + = v ( ) ( ) ( )21x ; f x f x a b ' + > e . Chng minh rngb a t > . Cho v d b a t = . Gii Ta c:( ) ( ) ( )( )( )( )221x ; 1 0x ;1f xf x f x a b a bf x'' + > e + > e+ ( ) ( ) ( ) ( ) arctan 0x ; arctan f x x a b f x x' + > e +tng trn( ) ; a bChuyn qua gii hn ta c: 2 2a b b at tt + s + > . V d:cot, a = 0 , b =y x t = . 3.25. Chof : R R l hm kh vi n cp hai sao cho ta c th tm c hm g :+ R R cho() () () () f x f x xg x f x x '' ' + = eR. Chng minh rng f(x) l hm b chn. Gii MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 78 Ta c:() () () () f x f x xg x f x x '' ' + = eR () () () () () () ( )22f x f x 2f x f x 2xg x f x ' ' '' ' + = . Xt hm s() () ( ) () ( ) ()2 2F x f x f x F x ' ' = + = () () ( )22xg x f x ' () F x 0' >vi x < 0v() F x 0 ' svi x 0 > . 3.26. Cho| | f : 0,1 R kh vi hai ln sao cho vi mi| | x 0,1 e ,() f x 1 '' s . Chng minh rng:( ) ( )1 1f 0 2f f 12 4| | + s |\ .. Gii Xt nh x sau: () () () () ()1 2xg x f x g x f x 1 0 g x2' ' '' '' = + = s l hm lm. T :( ) ( ) ( )1 1g g 0 g 12 2| |> + |\ .. Do :( ) ( ) ( )1 1 1f f 0 f 12 8 2| |+ > + |\ .( ) ( )1 1f 0 2f f 12 4| | + s |\ .. 3.27. Chof : R R kh vi hai ln, li sao cho() f x 0x > eR. Chng minh rngg : R R ( )xx2011x e f e . l hm li. Gii V f kh hai ln trnR nn g cng l hm kh vi hai ln trnR. Ta c:() ( ) ( )x 2010xx x2011 20111g x e f e e f e2011 ' ' = () ( ) ( ) ( ) ( )( ) ( ) ( )x 2010x 2010x 4021xx x x x2011 2011 2011 2011x 2010x 4021xx x x2011 2011 201121 1 2010g x e f e e f e e f e e f e2011 2011 20111 2009e f e e f e e f e 0x2011 2011 ('' ' ' '' = + + ( ' '' = + + > eR ( do f li v() f x 0x > eR) ------------------------------------------------------------------------------------------------ | |()( )g : 0,1x 1 x x f x2+RMATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 79 3.28. Cho( ) f xkh vi 2 ln tho( ) ( ) 0 1 0 = = f f , | | ()x 0;1minf x 1e= .Chng minh rng: | |( )0;1max 8e'' >xf x . Gii flintctrn| | | | ( )| | ()x 0;10;1 a 0;1 : f a minf x 1e - e = = .Suyrac( ) 0 ' = f a , ( ) 0;1 ae . Khai trin Taylor ti a:( )( ) ( )( )212u + = + f a x af x x a ,0 1 u < < . + Vi0 = x , ta c: ( )2 10 12''= +f ca, 10 < < c a+ Vi1 = x , ta c: ( )( )220 1 12''= + f ca, 21 < < a c . Do :( )1 228 '' = > f ca nu 1 2s a; ( )( )2 2281'' = >f ca nu 12> a . Vy | |( )0;1max 8e'' >xf x . 3.29.Gisfkhvilintcncphaitrn( ) 0;+ thomn () ()x xlimxf x 0 ,limxf x 0+ +'' = = . Chng minh rng:()xlimxf x 0+' = . Gii Vix 0 >ta c:( ) () () ( )1f x 1 f x f x f c2' '' + = + +vi( ) c x; x 1 e +Do :() ( ) ( ) () ( )x 1 xxf x x 1 f x 1 xf x . cf cx 1 2 c' '' = + + +. Suy ra:()xlimxf x 0+' = . 3.30.Chofkhvitrn| | a; b vgisrng( ) ( ) f a f b 0 ' ' = = .Chngminh rng nuf ''tn ti trong( ) a; bth tn ti( ) c a;b e sao cho: ( )( )( ) ( )24f c f b f ab a'' > Gii Ta c:( )()2f u a b b af f a2 2! 2''+ | | | |= + ||\ . \ . ; ( )( )2f v a b b af f b2 2! 2''+ | | | |= + ||\ . \ . BI TP V KHAI TRIN TAYLOR MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 80 vi a b a bu a;, v ;b2 2+ +| | | |e e ||\ . \ .. Do :( ) ( ) ( ) () ( ) (){ }2 2b a 1 b af b f a f v f u max f v , f u2 2 2 | | | |'' '' '' '' = s ||\ . \ .. t( ) ( ) (){ }f c max f v , f u '' '' '' = . T suy ra iu phi chng minh. 3.31. Cho f l hm lin tc kh vi cp hai trn( ) 1;1 v( ) f 0 0 = . Hy tnh gii hn sau:( )1xx 0j 1lim f jx+ ( ( =. Gii Theo cng thc Taylor, ta c: ( ) ( ) ( ) ( ) ()1 1x x2 2j 1 j 11 111 x xf jx f 0 jx f cjx j x f 0 x x2 2 (( (( = =| |(( + | ((| | \ .' '' ' = + = + |\ . vi() ( )1x2 2j 11x f cjx j x2, ( ( ='' =. Vf ''b chn trong ln cn ca 0 nn 1x2j 11 1 11 2 1x x xj6 ( ( =| || |(((+ + | | ((( \ .\ .= D thy()x 0lim x 0 += . T suy ra( )1xx 0j 1lim f jx+ ( ( =( ) f 02'= . 3.32. Gi s() f xl hm chn, kh vi hai ln v( ) f 0 0 '' = . Chng minh rng x 0 =l im cc tr. Gii () f xl hm chn ( ) () ( ) () ( ) f x f x f x f x f 0 0 ' ' ' = = = . Theo khai trin Taylor, ta c:() ( )( )( )2 2f 0f x f 0 x o x2''= + + . + Nu( ) f 0 0 '' >th ( )( )2 2f 0x o x 02''+ >vi x b suy rax 0 =l im cc tiu. +Tng t nu( ) f 0 0 '' > >vi mix 0 > . Chng minh rng nu () ()() ( )2xf x f xlim af x' '''='' th () ()() ( )2xf x f x 1lim2 af x''= '. Gii S dng quy tc LHospital ta c: ()()()() () ()() ( )2x x xf xxf x f x f x f xlim 1 lim lim axf x xf x '| | ' |'' | | ' ' '''\ . = = = |'' '''\ .. Do ()()xf xlim 1 axf x'= ''. T gi thit bi ton suy ra: a < 1.Ta c: ()()xxf x 1limf x 1 a''=' (*) Ta s chng minh()xlimf x= +. Theo cng thc Taylor, ta c: ( ) () () ( ) ( ) () ()2hf x h f x f x h f c , h >0 f x h f x f x h2' '' ' + = + + + > + . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 83 Choh ta c()xlimf x= +. Li p dng quy tc LHospital ta c: ()()() ()()x xxf x f x xf x 2 alim limf x f x 1 a ' ' '' + = =' . Kt hp vi (*) ta c:() ()() ( )()()()()2x xf x f x xf x f x 1 1 a 1lim lim . .f x xf x 1 a 2 a 2 af x '' ''= = =' ' '. 3.36. Chng minh rng vi f kh vi lin tc n cp 2 trnR tho mn ( ) f 0 1 = ,( ) f 0 0 ' =v( ) f 0 1 '' = th 2x20112x2011lim f ex| | |\ .+| || |= ||\ . \ .. Gii Ta c: x2011x ln fxx x2011lim f limex| | | | ||\ . \ .+ +| || |= | | |\ .\ .. ( ) ( ) ( )( )x t 0 t 0ln f 2011 t 2011f 2011 t2011limxlnf lim limt x2 t.f 2011 t+ ++ '| | = = |\ . ( )( ) ( )222t 02011 f 2011 t2011 2011lim2 22f 2011 t 2.2011 t.f 2011 t+''| |= = = |' \ . +. Vy x2011x ln fxx x2011lim f limex| | | | ||\ . \ .+ +| | | |= ||\ . \ .= 220112e| | |\ ..

3.37. Tm hm s() f xxc nh trnR tho mn iu kin:() ()2011f x f y x y s vi mix, y, x y e = R . Gii T gi thit ta suy ra: () ()()() ()()2010y xf y f x f y f x0 y x f x lim 0 f x C consty x y x ' s s = = = = . 3.38. Tm tt c cc hm f(x) xc nh v lin tc trnR sao cho ( ) ( ) 0x f x f x ' '' = eR. Gii S DNG O HM TRONG VIC GII PHNG TRNH HM MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 84 t( ) ( ) ( )2g x f x ' =( ) ( ) ( ) 2 0x g x f x f x ' ' '' = = eR ( ) ( ) ( ) g x C const f x const f x ax b ' = = = = + x eR. 3.39. Tmhm() f x 0 >v kh vi trn +Rng thi tho mn iu kin: () ()2 2 n nf x f y x yfx, y , n2 2-| |+ += e e | |\ .R N . Gii o hm 2 v ca ng thc cho ln lut theo bin x, bin y ta c: () ()() ()n n n 1n n 2 2f x f x x y nxf .2x y f x f y4 22 2| | '+' = | |+ +\ . () ()() ()n n n 1n n 2 2f y f y x y nyf .2x y f x f y4 22 2| | '+' = | |+ +\ . T suy ra: () () () () () ()()nn 1 n 1 n 1f x f x f y f y f x f x 2C f x Cxx y x n ' ' '= = = ( ) C 0 > . Th li thy ng. 3.40. Tm tt c cc hm s f(x) kh vi cp hai trnR v tho mn iu kin: () () f x f x '' =vi mix eR. Gii Gi s tn ti hm s f(x)tho mn yu cu bi ton. () () () () () () () () ( )xf x f x f x f x f x f x 0 e f x f x 0''' ' ' '' '( = + = =(( () () () ( ) ()x x 2x x 2xCf x f x Ce e f x C.e e f x .e B2 '' = = = +t CA2= , ta suy ra:()x xf x Ae Be ,A ,B: const= + . Vy()x xf x Ae Be ,A ,B: const= +l hm s cn tm. C-MT S BI TP NGH MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 85 3.41. Xem xt tnh kh vi ca hm s sau()22 xx e,x 1f x1,x 1es= >. 3.42. Chng minh rng hm s()2x cos ,x 0f x x0 , x = 0t == khng kh vi ti cc im n2x, n2n 1= e+Z nhng kh vi ti 0 l im gii hn ca dy( )nx . 3.43. Cho f kh vi ti 0x . Hy tnh cc gii hn sau: a) () ( )() ( )( )0x00x x0f x e f xlim , x = 0 , f 0 0.f x cos x f x' = b)( )n0 0 2xk 1klim f x nf xn=| || |+ ||\ . \ . 3.44. Cho ( )2n2nf ln 1 x, n = + eN. Hy chng minh rng: ( )( )2n2nf 1 0. =3.45. Xt 0 1 2011b , b ,..., b eR tho mn: 2 2010 20112 2010 20110 12 b 2 b 2 bb 2b ... 03 2011 2012+ + + + + = . Chng minh rng phng trnh:2011 22011 2 1 0b ln x ... b ln x b ln x b 0 + + + + =c t nht mt nghim trong ( )21;e . 3.46. Cho cc hm s, , | lin tc trn| | a; bv kh vi trn( ) a; b . Xt()() () ()( ) ( ) ( )( ) ( ) ( )x x xx det a a ab b b | | | | | |u = | |\ ..Chng minh rng tn ticsao cho( ) c 0 ' u = . 3.47. Cho f, g l cc hm s kh vi lin tc n cp n ti mt ln cn ca im a tho mn:( ) ( ) ( ) ( )( )( )( )( )n 1 n 1f a g a ,f a g a, ...,f a g a ' ' = = = =v ()( )( )( )n nf a g a = . Tnh () ( )() ()f x g xx ae elimf x g x. 3.48.Cho1 > . Ta k hiu() f l mt nghim thc ca phng trnh: ( ) x 1 lnx + = . Chng minh rng: () flim 1/ ln = . 3.49. Hy chng minh cc bt ng thc sau: MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 86 a)( ) ( ) ( )e x e xe x e x , x 0;e ++ > eb) ( )2 232 xx 4x sin x , x 0;2ttt t (+ s e ( . c) ( )k2011x xk 0x xe e 1k! 2011 = < ,( ) x 0; e + . d) ( )2011kk 12011 12011 a3k kk 13a ea 0 , k = 1, 2011e==| |s > |\ .[. e)( ) ( )20112 2011 kk k2011k 02011k 2011x C x 1 x4= s. f) 2 222 2lnab b a a b a babba ba + +< < < 0,a b a = . 3.50.Chng minh rng nu cc o hm() () f x, f x '' '''tn ti th a)()( ) () ( )( )2x 0f x x 2f x f x xf x limxA + A + A'' =A. b)()( ) ( ) ()( )2x 0f x 2 x 2f x x f xf x limxA + A + A +'' =A.c)()( ) ( ) ( ) ()( )3x 0f x 3 x 3f x 2 x 3f x x f xf x limxA + A + A + + A ''' =A. 3.51. Cho: f R R kh vi n cp n + 1 trnR. Chng minh rng vi mi xeR tn ti( ) 0,1 cesao cho : a)( ) ( ) ( ) ( )20 ...2xf x f xf x f x ' '' = + + + ( )( )( ) ( )( )( )( )11 211 1! 1 !n nn nn nx xf x f cxn n++ +++ + +. b)( ) ( ) ( )( )( )( )2 2... 11 1 !1nnnnf x x x xf f x f xx x nx| |' = + + + |+ +\ . + +( )( )( )( )212 21111 ,x 11 !1nnnnx cxfx xnx++++| | + |+\ . = ++. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 87 6.52. Cho Q(x) l mt a thc bc n. Chng minh rng: ( )( )( )( )( )( )( )1 10 0011 ! 1 !i in nii ii iQ Q xx xi i+ += == + + . 6.53. Cho I, J l hai khong m v: f J R ,: f I J l cc hm kh vi v hn trn J, I. Chng minh rng: ( )( )( )( )( ) ( )( ) ( )( )( )1 201!...1! 2! !!nkk knnknjjg t g t g tnf g t f g tnk=| |' '' | | | |= | || |\ . \ .\ .[ Trong 1

njjk k== v ly tng trn cc gi tr jksao cho: 1njjjk n==. 6.54. Cho: f R R kh vi n cp2 1 n +trnR. Chng minh rng vi mi xeR, tn ti( ) 0,1 u esao cho: ( ) ( )32 20 . . ...1! 2 2 3! 2 2x x x xf x f f f| | | | | |' ''' = + + + |||\ . \ . \ .+ ( )( )( )( )( )2 1 2 12 1 2 12 2. .2 1 ! 2 2 2 1 ! 2n nn nx x xf f xn nu + + | | | | | |+ + ||| +\ . \ . \ .. 6.55. Gi s| | : , f t t R kh vi cp hai trn| | , t t v t | |( )( ),sup, i = 0,1,2.iix t tK f xe =Chng minh rng: a)( ) ( ) | |2 2 0 2 ,2K Kf x x t x t tt c' s + + e . b) 1 0 22 K K K svi 02KtK> . 6.56. Cho f kh vi n cp hai trnR , t ( )( )0,supixK f xe += < +vi( ) 1, 2,...,2 k j j = > . Chng minh rng: ( )120 22 , i = 1,2,...,j 1i i i j ij jiK K Ks . 6.57. Gi s f kh vi lin tc n cp n trnR, 0x eR. Chng minh rng: ( )( ) ( ) ( )( ) 0 0001lim 1nn kn kn nkf x C f x kqqq== +. 6.58. Chng minh rng nu| | ( ), f C a b evf'tn ti trn( ) , a bth MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 88 ( )( )( ) ( )( )( ),,inf supx a bx a bf b f af x f xb a ee' ' s s. 6.59. Gi s rng hm f lm v tng thc s trn( ) , a bvi{ } ,b a e R . Chng minh rng nu( ) ( ) , f x a x evi( ) , x a b ev( ) lim 1x af x++' =th ( ) ,y a,b x eta c: ( ) ( )() ()11lim 1n nn nnf x f xf y f y++= y0 0 0...nnf f f f =_. 6.60. Gi s| | ( )2, f C a b e,() ( ) 0 f a f b eth ()baf x dx 0 >}. 6.2. Mnh 2 Nu f, g l cc hm s lin tc trn| | a, bv() () | | f x g x x a, b s eth () ()b ba af x dx g x dx s} }. 6.3. Mnh 3 Nu f l hm s lin tc trn| | a, b ,() | | f x 0x a, b > ev f(x) khng ng nht bng 0 trn| | a, bth()baf x dx 0 >}. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 94 6.4. Mnh 4 Nu f, g l cc hm s lin tc trn| | a, bv() () | | f x g x x a, b s ev () () f x ,g xkhng ng nht vi nhau trn| | a, bth() ()b ba af x dx g x dx 1 vf, g l cc hm s lin tc trn| | a, b . Khi : () () () ()1 1 1b b bp p pp p pa a af x g x dx f x dx g x dx| | | | | |+ s + |||\ . \ . \ .} } }. 6.10. Mnh 10 ( Bt ng thc Holder) Cho p, q > 1 tho1 11p q+ =v f, g l cc hm s lin tc trn| | a, b .Khi :()() () ()1 1b b bp qp qa a af x g x dx f x g x| | | |s + ||\ . \ .} } }. III. TCH PHN SUY RNG TRN KHONG V HN 1. nh ngha Cho hm s| ) : , f a + R kh tch trn mi on| | , a A ( A > a). Biu thc:( ) ( ) limAAa af x dx f x dx++=} } (1)c goi l tch phn suy rng ( loi 1) ca hm f(x) trong khong| ) , a + . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 95 Nu gii hn (1) tn ti v hu hn th tch phn( )af x dx+} c gi l hi t. Nu gii hn (1) khng tn ti hoc bng th tch phn( )af x dx+} c gi l phn k. Tch phn( )af x dx+} c nh ngha tng t. Nu( ) : , f + R l hm kh tch trn mi on hu hn | | ( ) , , B A c +th biu thc ( ) ( )limAABBf x dx f x dx++=} } (2) c gi l tch phn suy rng ca hm ( ) f xtrong khong( ) ; + . Nu gii hn (2) tn ti hu hn th tch phn( ) f x dx+} c gi l hi t; trong trng hp ngc li ta ni tch phn ny phn k. Cho a l s thc bt k. Nu c hai tch phn( ) ()+a ,f xaf x dx dx} } cng hi t th( ) ( ) ( )aaf x dx f x dx f x dx+ + = +} } }. Nu tch phn suy rng trn cc khong( | | ) ,,a,+ a ,( ) , +ca hm f(x) hi t th ta ni f(x) kh tch trn cc khong tng ng. 2. Tiu chun hi t Cauchy Tch phn( )af x dx+} hi t( )0 00, A,A, A > A :AAf x dx c c'''' > - a a Asao cho :( ) ( ) 0 f x g x s svi mi| ) , x a e + . Khi : nu( )ag x dx+} hi t th( )af x dx+} hi t. Nu ( )af x dx+} phn k th ( )ag x dx+} cng phn k. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 96 b) Gi s f(x) v g(x) xc nh v khng m trong khong| ) , a + , kh tch trong mi on hu hn| |( ) ,A > a a Asao cho tn ti gii hn : ( )( )lim , 0 < k < + .xf xkg x+= Khi cc tch phn( )af x dx+} v( )ag x dx+} cng hi t hay cng phn k. c) Gi s( ) f xc dng :( )( )( )> 0xf xxoo =Khi :Nu1 o >v( ) x l hm khng m v b chn trn : ( ) | ) 0 x a ,+ x M s s e th tch phn( )af x dx+} hi t. Nu1 o s , cn( ) x l hm khng m v b chn di : ( ) | ) 0 x a,+ m x < s e th tch phn( )af x dx+} phn k. 4. Cc nh l Abel v Dirichlet 4.1. nh l Abel Gi s f(x) v g(x) xc nh trong khong| ) , a + . Gi s rng : Tch phn ( )af x dx+} hi t ; Hm g(x) n iu v b chn trong| ) , : a + ( ) | ) x a,+ , g x L s e L l hng s. Khi :( ) ( )af x g x+} hi t. 4.2. nh l Dirichlet Cho cc hm s f(x) v g(x) xc nh trong khong| ) , a + . Gi s rng : a) f(x) kh tch trn on hu hn| | ( ) , A > a a Asao cho : ( )A a , KAaf x dx K s >} l hng s. b) Hm g(x) n iu dn v 0 khix + :( ) lim 0xg x+= . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 97 Khi :( ) ( )af x g x+} hi t. 5. S hi t tuyt i v bn hi t Cho hm f(x) xc nh trong khong| ) , a + . Nu( )af x dx+} hi t th tch phn( )af x dx+} cng hi t. Khi tch phn( )af x dx+} c gi l hi t tuyt i. Nu tch phn( )af x dx+} hi t nhng tch phn( )af x dx+} phn k th tch phn( )af x dx+} c gi l bn hi t hay hi t khng tuyt i. B.CC DNG BI TP dng bi tp ny c gi cn ch nhiu hn n cc bi ton vn dng tch phn xc nh tnh gii hn ca dy s. Thc t c rt nhiu bi ton dy s m ch s dng nhng kin thc trong ni b dy s th khng th gii quyt c hoc nu gii quyt c th tn km nhiu thi gian v cng sc. V vy chng ta cn linh hot trong cng viclm xut hin tng tch phn trong bi ton gii hn dy s. yl mt trong nhng dng ton hay thng xuyn c mt trong cc thi Olympic ton sinh vin ton quc cu v gii hn dy s. 4.1. Cho f l hm lin tc, dng trn on| | 0,1 . Chng minh rng: ()10lnf x dxnn1 2 nlim f .f ...f en n n+}| | | | | | = |||\ . \ . \ .. Gii Ta c: nni 11 i 1 2 nlnf ln f .f ...fn n n n nn1 2 nf .f ...f e en n n=| | | | | | | | ||||\ . \ . \ . \ .| | | | | | = = |||\ . \ . \ .. BI TP VNH NGHA TCH PHN XC NH MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 98 ni 11ilnfnn=| | |\ .l tng tch phn ca hm() () g x lnf x =trn on| | 0,1ng vi cch chia on| | 0,1thnh n phn bng nhau v chn | | ( )i i i 1 i1x x , x i 1, 2,..., nn= = e = . V f l hm lin tc, dng trn on| | 0,1nn g(x)l hm xc nh, lin tc trn on . Do :()1nni 101 ilnf x dx lim lnfn n+=| |= |\ .}. Vy ()10lnf x dxnn1 2 nlim f .f ...f en n n+}| | | | | | = |||\ . \ . \ .. 4.2. Chng minh rng gii hn n2 nsin sin sinn 1 n 1 n 1lim ... 01 2 nt t t| | |+ + ++ + + > | |\ . Gii Xt hm s()( |sin x, x 0,f xx1 , x = 0te= .R rng f(x) l mt hm lin tc trn| | 0,tv dng trn| ) 0,t . Nhng vy f(x) kh tch Riemann v()0f x dx 0t>}. Ta c: ()nn ni 102 n isin sin sin sinn 1 n 1 n 1 n 1lim ... lim f x dx 0i1 2 n n 1n 1tt t t ttt =| | |+ + + ++ + + = = > |+ |\ . +}. 4.3. Tnh n1 1 1lim ...2 8 6n 4n n n3 3 3+| | |+ + + | | + + +\ .. Gii t ni 11 1 1 1 2 1... .2 8 6n 4 6i 42 nn n n 13 3 3 3n=+ + + = + + + + MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 99 Xt hm s()1f x1 x=+lin tc trn| | 0, 2 nn kh tch trn on ny. Chia| | 0, 2bi cc im chia( )i2ixi 0,1, 2,..., nn= =Chn| |i i 1 i i 1 i2 1 6i 4x x x , x3 3 3n = + = eD thy

n1 1 1lim ...2 8 6n 4n n n3 3 3+| | |+ + + | | + + +\ . ( )( )2ni i i 1ni 101 1 dxlim . f x x ln 32 2 1 x+== = =+} 4.4.Chng minh rng nu f kh vi lin tc trn| | 0,1th ()( ) ( )1nni 10f 1 f 0 1 ilimn f f x dxn n 2= | || | = ||\ . \ .} Gii Trc ht, chng ta c:() ()i1 n n n ni 1 i 1 i 1 i 1 0n1 i 1 in f f x dx n f f x dxn n n n = = = | || | | | | | | = ||| |\ . \ . \ .\ . } } =() () ( )i in n n nii 1 i 1 i 1 i 1n ni in f f x dx n f x x dxn n= = | || | | |' = |||\ . \ . \ . } } t()ii 1 ix ,n nm inf f x(e ( ' = ; ()ii 1 ix ,n nM sup f x(e ( ' =Do :() ( )i i in n ni i ii 1 i 1 i 1n n ni i im x dx f x x dx M x dxn n n | | | | | |' s s |||\ . \ . \ .} } } Suy ra:() ( )in n ni i ii 1 i 1 i 1n1 i 1m n f x x dx M2n n 2n= = | |' s s |\ . } Hn na,f 'l mt hm lin tc nn () ()( ) ( )1 1nni 10 0f 1 f 0 1 i 1limn f f x dx f x dxn n 2 2= | || |' = = ||\ . \ .} }. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 100 4.5. Chng minh rng() | | f x x =kh tch trn| | 0, 2011v tnh | |20110x dx}. Gii R rng() | | f x x =b chn trn| | 0, 2011v 2011 im gin on: 1,2,,2011 Ta c: | | | |2011 i 12011 2010i 0 i 00 ix dx x dx i 2021055+= == = = } }. 4.6. Gi s f(x) kh tch trn on| | 0,1v()10f x dx 0 >}. Chng minh rng tn ti on| | | | a, b 0,1 cm trong on () f x 0 > . Gii Chia u on| | 0,1bi n im chia( )0 nix 0 , xi 1, 2,..., nn= = = . Chn| |i i 1 ix , x e , ta lp c tng tch phn( ) ( )nn ii 11f , fno ==. V f(x) kh tch trn on| | 0,1nn( ) ( ) ()1nn in ni 101lim f , lim f f x dxno + +== =}. Gi s trn mi on con| | | | a, b 0,1 c , hm f(x) c cha nhng im x lm cho() f x 0 s . Khi ta d dng suy ra c( )if 0 s . Do : ( ) ()1nn0lim f , f x dx 0 o += s}. iu ny mu thun vi gi thit. Vy ta c c iu phi chng minh cho bi ton ny. 4.7. Cho hm f(x) xc nh trn| | a.b . a) Nu() f xl hm kh tch trn on| | a, bth hm f(x) c kh tch trn on hay khng? b) Nu()2012f xl hm kh tch trn on| | a, bth hm f(x) c kh tch trn on hay khng? Gii Cha hn l() f xkh tch trn| | a, b ! Xt v d sau y: Cho()1 ,xf x1, x \e= eR . Hm ny khng kh tch trn mi on | | a, b . Th nhng() ()2012f x , f xkh tch trn on| | a, b . BI TP V S KH TCH CA HM S MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 101 Ta c: a)()b ba af x dx 1dx b a = = } } b)()b b2012a af x dx 1dx b a = = } }. 4.8. Cho() | |2p p p 1; x ,, p = 0, n 1f x; x 0,1n n n1 ; x = 1+| ||e | |

= e\ . . vneN. Chng minh rng() f xkh tch trn| | 0,1v tnh()10f x dx}. Gii Hm() f xb chn v gin on ti cc im( )kkxk 1, 2,..., nn= = . Do f(x) kh tch trn| | 0,1 . Ta c:() ()( )( )p 12 1 n n 1 n 1 n 122 3 2p 0 p p 0 p 0nn 1 2n 1 p 1 1 1f x dx f x dx . p .n n n 6 n+ = = = | |= = = = |\ . } } 4.9.Cho f l mt hm lin tc trn| | ; a bv( ) 0baf x dx =}. Chng minh rng tn ti( ) ( ) ( ) ; :cac a b f x dx f c e =}. Gii Xt hm:( ) ( )xxag x e f t dt=} g lin tc trn| | ; a b , kh vi trn( ) ; a b( ) ( ) 0 g a g b = = .Theo nh l Rolle tn ti( ) ( ) ; : 0 c a b g c ' e = . M( ) ( ) ( )xxag x e f x f t dt | |' = |\ .}, v th( ) ( ) ( )c ca af c f t dt f x dx = =} }. 4.10.Gi s f, g | | ( ); C a b e . Chng minh rng tn ti( ) ; c a b esao cho( ) ( ) ( ) ( )b ba ag c f x dx f c f x dx =} }. BI TP XOAY QUANH CCNH L GI TR TRUNG BNHMATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 102 Gii Xt( ) ( ) ( ) ( ) , Gx xa aF x f t dt x g t dt = =} } Suy ra:( ) ( ) F x f x ' =,( ) ( ) G x g x ' =p dng nh l Cauhy ta c: - ce( ) ; a b : ( ) ( )( ) ( )( )( )F b F a F cG b G a G c' =' - ce( ) ; a b : ( )( )( )( )babaf t dtf cg cg t dt=}} - ce( ) ; a b :( ) ( ) ( ) ( )b ba ag c f x dx f c f x dx =} }. 4.11.Gi s f, g | | ( ); C a b e . Chng minh rng tn ti( ) ; c a b esao cho( ) ( ) ( ) ( )c ba cg c f x dx f c f x dx =} }. Gii Xt hm:( ) ( ) ( )x ba xF x f t dt g t dt =} } Flin tc trn| | ; a b , kh vi trn( ) ; a bv( ) ( ) F a F b = . V th theo nh l Rolle ta c:( ) ( ) ; : 0 c a b F c ' - e =M( ) ( ) ( ) ( ) ( )b xx aF x f x g t dt g x f t dt ' = } } Do :( ) ; : c a b - e ( ) ( ) ( ) ( )c ba cg c f x dx f c f x dx =} }. 4.12.Cho| | ( )20;1 f C e . Chng minh rng tn ti( ) 0;1 cesao cho: ( ) ( ) ( ) ( )101 10 02 6f x dx f f f c ' '' = + +}. Gii Ta c:( ) ( )( ) ( ) ( ) ( ) ( )1 1 1100 0 01 1 1 f x dx f x d x x f x x f x dx ' = = } } } ( )( )( )( )( )12 21001 102 2x xf f x f x dx ' '' = +}. p dng nh l gi tr trung bnh ca tch phn: MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 103 tn ti( )( )( ) ( ) ( ) ( )21 120 01 1 10;1 : 12 2 6xc f x dx f c x dx f c'' '' '' e = =} }. Do tn ti( ) 0;1 cesao cho:( ) ( ) ( ) ( )101 10 02 6f x dx f f f c ' '' = + +}. 4.13.Cho f lin tc trn| | ; a b. t( )1bac f x dxb a= }. Chng minh rng: ( ) ( )2 2 b ba af x c dx f x t dx t s e} }R. Gii Xt( ) ( ) ( ) ( )22 22b b ba a ag t x t dt b a t f x dx t f x dx| |= = + |\ .} } }. g(t) l tam thc bc hai theo t, g(t) t cc tiu ti( )01bat f x dx cb a= = }. Vy( ) ( )2 2 b ba af x c dx f x t dx t s e} }R. 4.14.Chng minh rng nu f kh tch Riemanntrn| | ; a bth ( ) ( ) ( ) ( )2 22sin cosb b ba a af x xdx f x xdx b a f x dx| | | |+ s ||\ . \ .} } }. Gii p dng bt ng thc Schwarz, ta c: ( ) ( )( ) ( ) ( ) ( )2 22 2 2 2 2sin cossin cosb ba ab b b b ba a a a af x xdx f x xdxf x dx xdx f x dx xdx b a f x dx| | | |+ s ||\ . \ .s + = } }} } } } } 4.15.Chng minh rng nu f dng v kh tch Riemann trn| | ; a bth ( ) ( )( )2b ba adxb a f x dxf x s} }.BI TP V BT NG THC TCH PHN MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 104 Hn na nu( ) 0 m f x M < s sth( )( )( ) ( )224b ba am M dxf x dx b af x mM+s } }. Gii + p dng bt ng thc Cauchy Schwarz, ta c: ( ) ( )( )( )( )22 1.b b ba a adxb a f x dx f x dxf xf x| | | = s |\ .} } }. + V( ) 0 m f x M < s snn ( ) ( ) ( ) ( )( )0 , a x bf x m f x Mf x s s sTa c: ( ) ( ) ( ) ( )( )( ) ( )( )00b b b ba a a af x m f x Mdxdx f x dx m M dx mMf x f x s + + s} } } } ( )( )( )( )( )( )( ) ( ) .b b b ba a a adx dxf x dx mM m M b a mM m M b a f x dxf x f x + s + s + } } } }Do :( )( )( )( ) ( ) ( )2b b b ba a a adxmM f x dx m M b a f x dx f x dxf x| |s + |\ .} } } } Xt hm s:( )2y g t t kt = = + .Hm s t cc i ti 2kt =vi gi tr cc i l 24k. Vi( )( ) ( ) , t = bak m M b a f x dx = + } ta c: ( )( ) ( ) ( )( ) ( )2 2 24b ba am M b am M b a f x dx f x dx+ | |+ s |\ .} }. Do : ( )( )b ba adxmM f x dxf x} }( ) ( )2 24m M b a + s ( )( )b ba adxf x dxf x} }( ) ( )2 24m M b amM+ s . 4.16.Cho| ) : 0; f + Rl mt hm lin tc kh vi. Chng minh rng: ( ) ( ) ( )| |( ) ( )21 1 13 20,10 0 00 maxxf x dx f f x dx f x f x dxe| |' s |\ .} } }. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 105 Gii t | |( )0,1maxxM f xe' = . Khi ( ) | | ( ) | | x 0;1 x 0;1 f x M M f x M s e s s e . Nhn( ) 0 f x >vo tng v ca bt ng thc ny ta c : ( ) ( ) ( ) ( ) Mf x f x f x Mf x ' s s,| | 0;1 xeSuy ra:( ) ( ) ( ) ( )0 0 0x x xM f t dt f t f t dt M f t dt ' s s} } } ( ) ( ) ( ) ( )2 20 01 102 2x xM f t dt f x f M f t dt s s} }. n y ta tip tc nhn ( ) 0 f x >vo tng v ca bt ng thc ny c: ( ) ( ) ( ) ( ) ( ) ( ) ( )3 20 01 102 2x xMf x f t dt f x f f x Mf x f t dt s s} },| | 0;1 xe . Ly tch phn 2 v trn| | 0;1ca bt ng thc ny: ( )210M f x dx| | s |\ .}( ) ( ) ( ) ( )21 1 13 20 0 00 f x dx f f x dx M f x dx| | s |\ .} } } ( ) ( ) ( ) ( )21 1 12 20 0 00 f x dx f f x dx M f x dx| | s |\ .} } } hay( ) ( ) ( )| |( ) ( )1 1 13 20,10 0 00 maxxf x dx f f x dx f x f x dxe| |' s |\ .} } }. 4.17.Tm ( ) ( )12 20min 1fK x f x dxe= +} , y| | ( ) ( )100,1 : 1 f C f x dx = e = ` )}. Giip dng bt ng thc Schwarz ta c: ( ) ( ) ( ) ( )221 1 1 12 2 2220 0 0 011 1 1 .1 41dxf x dx x f x dx x f x dx Kxxt | || |= = + s + = ||+\ . +\ .} } } } Suy ra: ( ) ( )12 204min 1fK x f x dxte= + >}. 4.18.Cho| | ( ) ( ) ( )0 00;1 : sin cos 1 M f C f x xdx f x xdxt t = e = = ` )} }. Tm( )20minf Mf x dxte }. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 106 Gii Cho( ) ( )02sin cos f x x xt= + . +R rng 0f M e . + i vi hm bt kf M e ,( ) ( )2000 f x f x dxt >( }. Suy ra:( ) ( ) ( ) ( ) ( )2 2 20 0 00 0 0 08 4 42 f x dx f x f x dx f x dx f x dxt t t tt t t> = = =} } } }. Vy cc tiu t c khi 0f f = . 4.19. Chng minh rng: 1201101x 1 xdx2012 2013 s}. Gii Ta c:( ) ( )1 11 1 12 22011 2011 2011 20110 0 0x . x 1 x dx x dx x 1 x dx| | | | s ||\ . \ .} } }== 1 11 12 22011 20120 01 1 1x dx x dx2012 2013 2012 20121 | | | | = = ||\ . \ .} }12012 2013. 4.20.Tmgitrlnnhtca ( )( )201110120110| | |\ .= }}f x dxSf x dxviflintc,dngtrn | | 0;1 . Gii p dng bt ng thc Holder ta c: ( ) ( ) ( )2010 1 12011 2 2 2 22011 2011 20112011 20112010 201120100 0 0 0.1 1 2| | | | | |s = |||\ . \ . \ .} } } }f x dx f x dx dx f x dx( ) ( )( )( )2011220112 22010 2011 0220110 002 2| | || |\ . s = s |\ .}} }}f x dxf x dx f x dx Sf x dx.Vy 2010max 2 = S . 4.21.Chngminhrngnu| | ( )f C a, b e ldngvlmtrn| | a, b th () ( )| | ()bx a,ba1f x dx b a maxf x2e> }. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 107 Gii f lin tc trn| | 0;1nn| | ( )| | ()x a ,bc a, b : f c maxf xe- e = . Ta c:() () ()b c ba a cf x dx f x dx f x dx = +} } } =( ) ( ) ( ) ( )1 10 0c a f 1 x a xc dx b c f 1 x c xb dx + + +(( } } ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 10 0c a 1 x f a xf c dx b c f 1 x f c xf b dx > + + + =(( } } =( ) ( ) ( )( ) ( ) ( )( ) ( )f a f c f b f c 1c a b c b a f c2 2 2+ + + > . Do chng ta c iu phi chng minh. 4.22.Cho| | ( ) () ()1 10 0f 0,1 : f x dx 3, xf x dx 2 = e9 = = ` )} }. Tm()12f0min f x dxe }. Cho v d v mt hm s tho mn nhng yu cu nh th. Gii p dng bt ng thc Cauchy-Schwarz, ta c: ( ) ()( ) () ( )21 1 12 220 0 02 3t f x x t dx f x dx x t dx| |+ = + s + |\ .} } } ()( )( ) () ( )21 12 22t0 03 2 3tf x dx t f x dx max t3t 3t 1 e+ > = >+ +} }R. Kho st hm( ) t , da vo bng bin thin ta d dng suy ta c ( )tmax t 12 e=R. Vy()12f0min f x dx 12e=}. Chng hn ta xt hm() f x 6x = . 4.23.Cho| ) | ) f ,g: 0, 0, + +l hai hm lin tc, khng m . Gi s () ( )( )x0f x 2011 f t g t dt x > 0. s + } Chng minh rng ()( )x0g t dtf x 2011ex > 0.}s Gii Ta s chng minh( )( )x02011 f t g t dt + s}( )x0g t dt2011e } MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 108 Tht vy! ( )( )x02011 f t g t dt + s}( )x0g t dt2011e }( )( ) ( )x x0 0ln 2011 f t g t dt ln2011 g t dt ( + s ( } }()()( )( )()x xu0 00f u g udu g u du2011 f t g t dt s+} }}. iu ny hon ton ng v n c suy ra t gi thit ca bi ton. 4.24. Gi s| ) ( )f C 0, e +v t( ) ( )1n0x f n x dx,n 0,1, 2,... = + =}. Hy tm ( )1n0lim f nx dx} bit nnlimx 2011= . Gii Ta c:( ) () ()1 n k 1n 1n x xk 00 0 k1 1lim f nx dx lim f x dx lim f x dxn n+ == =} } } ( )n 1n 1 kk 0n nk 0x1lim f x k dx lim 2011n n= == + = = ( V nnlimx 2011= ). 4.25. Choflin tc trnR. Tm( ) ( ) ( )01limbhaf x h f x dxh+ }. GiiTa c:( ) ( ) ( ) ( ) ( )b b h ba a h af x h f x dx f x dx f x dx+++ = } } }( ) ( ) ( ) ( )b b h a h ba h b a a hf x dx f x dx f x dx f x dx+ ++ += + } } } } , | | , 0,1 u u' e . BI TP V GII HN CA TCH PHN( ) ( ) ( ) ( )a b ha h bf x dx f x dx hf a h hf b h u u++' = + = + + +} }MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 109 ( ) ( ) ( ) ( ) ( )01limbhaf x h f x dx f b f ah + = }. 4.26. Tnh 2n35nnxdxlim nx 1| | |+\ .}. Gii Ta c nh gi sau: ( )2n 2n 2n3 3 345 4n n ndx xdx dxn n nx 1 x x 1< e ( . Khi ta c:( )22 2k tksin tk0 010 e dt e dt 1 0k2k et ttt| |s s = |\ .} }. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 110 Vy 2ksin tk0lim e dt 0t=}. 4.29. Tnh 2010x20102011x 001lim sin tdtx+}. Gii p dng quy tc LHospital , ta c: 20102010x20102009 x2010 02011 2011 2010x 0 x 0 x 0 x 00sin tdt1 2010x sin x 2010 sin xlim sin tdt lim lim lim 1x x 2011x 2011 x+ + + + = = = =}} 4.30. Tnh x2011 2011 x01 2010lim dtln xt 2012| | |+\ .}.Gii p dng quy tc LHospital , ta c: xx2011 201102011 2011 2011 2011 x x x02010dt1 2010 2010xt 2012lim dt lim lim 2010.ln x ln xt 2012 x 2012+ + +| |+= = = |+ +\ .}} 4.31. Tnh 2nx2xnelim dx1 e+ + Gii Ta c:| |2nx 2nx 2nx2x 2x 2x2x 2xe e ee 1x 0,1 2e 1 e 22e 1 e 2 > e > + > s s+ 2nx 2nx 2nx 2n 2 2nx 2nx 1 1 1 12x 2x 2x0 0 0 0e e e 1 1 e e 1 1 edx dx dx . dx .2e 1 e 2 4 n 1 1 e 4 n s s s s+ + +} } } } n y ta s dng nh l kp v dng suy ra c 2nx2xnelim dx=01 e+ +. 4.32. Tnh n nn0lim2 cos xcos nxdxt}. Gii t nn0I cos xcos nxdxt=}. Ta c:( ) ( )n n nn00 01 1I cos xd sin nx cos xsin nx sin nxd cos xn nt tt= = } } ( ) ( )n 1 n 10 01sin nxsin xcos xdx cos n 1 x cos n 1 x cos xdx2t t = = +( } } MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 111 | |n 1n 11 1I cos x cosnxcos x sin nxsin x dx2 2= ( )n 1 n n n 1 n 2 0 2 n n1 1 1 1 1I I I I I I2 2 2 2 2 2t = = = = = . Suy ra n nn0lim2 cos xcos nxdxt} = t . 4.33. Tnh n2 n nxI dxx x1 x ...2! n!=+ + + +}( *n eN ). Gii t() ()( )2 n n 1n nx x xf x 1 x ... f x 1 x ...2! n! n 1 !' = + + + + = + + +. Ta c: () () ( )()()()()n n nn nn nn! f x f x f xI dx n! 1 dx n!x n!lnf x Cf x f x' | | '= = = + |\ .} } 2 nx xn!x n!ln 1 x ... C2! n!| |= + + + + + |\ . 4.34. Tnh x bb a xae eI dxx=}. Gii t 2 2ab ab abt dt dx dx dtx x t= = = . i cnx a t b, x = b t = a = = . Ta c: x b t t b bb a b a t a a t x2a b ae e e e ab e eI dx . dt dt I 2I 0 I 0abx t tt = = = = = =} } }. 4.35. Tnh tch phn ( )( ) ( )42ln 9 xI dxln 9 x ln x 3= + +}. MT S BI TP TNH TONCC DNG TCH PHN MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 112 Gii tx 6 t = , ta thu c:( )( ) ( )42ln t 3I dtln t 3 ln 9 t+=+ + }. Suy ra: 422I dx I 1 = =}. 4.36. Tnh tch phn 1 2xx121I 1 x e dxx+| |= + |\ .}. Gii Ta c: 1 2xx121I 1 x e dxx+| |= + |\ .}1 1xx1211 x e dxx+| |= + |\ .}1 2xx111 x e dxx+| |+ + |\ .} = J K + . t 21 t t 4t x xx 2 = + =+ tnh J, ta i bin 22t t 4 1 tx dx 1 dt2 2t 4| | = = |\ .. Thay vo v bin i n gin cho n khi c kt qu nh sau: 522 t221 tJ 1 t t 4 e dt2t 4| |= + + |\ .}. + tnh K, ta i bin 2t t 4x+ =v thc hin tnh ton nh tnh ton tm J, ta thu c kt qu 522 t221 tK 1 t t 4 e dt2t 4| |= + + + |\ .}. + T suy ra: 522 t22tI J K t 4 e dtt 4| |= + = + |\ .}. + n y th bi ton ht sc n gin, ta s dng tch phn tng phn v tnh ton, thu gn c kt qu cui cng 23I e e2= . 4.37. Tnh tch phn: ( ) ( )22 2 2011 20110I cos cos x sin sin x dxt (= + }. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 113 Gii tx t2t= , ta thu c tch phn sau y: ( ) ( ) ( ) ( )2 22 2 2 2 2011 2011 2011 20110 0I cos sin t sin cos t dt cos sin x sin cos x dxt t ((= + = + } }. Suy ra: 202I 2 dx I2tt= =}. 4.38. Cho m, n l cc s nguyn dng v a < b, hy tnh ( )( )! !m nbab x x adxm n }. Vn dng cng thc va tnh tnh ( )1201nx dx }. Gii Dng cng thc tch phn tng phn ta c: ( )( ) ( ) ( )( )1! ! ! 1 !m n m nb ba ab x x a b x x adx dm n m n+| | =| |+\ .} } ( )( )( )( )( )( )( )( )( )( )( )1 1 1 1 1! 1 ! 1 ! 1 ! 1 ! 1 !bm n m n m nb ba aab x x a b x x a b x x adx dxm n m n m n+ + + = + =+ + +} }. Dng cng thc tch phn tng phn nhiu ln ta c: ( )( ) ( )( )( )( )( )( )1 1! ! ! 1 ! 1 !bm n n m n m n mb ba aab x x a x a x a b adx dxm n n m n m n m+ ++ ++ = = =+ + + + +} }. p dng vo bi ton c th nh sau: ( ) ( ) ( ) ( ) ( ) ( ) ( )1 12 1220 11 1 2 2.4.6...21 1 1 !2 2 2 1 ! 1.3.5. 2 1nn n n nx dx x x dx nn n+ = = =+ +} }. 4.38. Tn ti hay khng hm kh vi lin tcftha mn iu kin ( ) ( ) ( ) 2 , f f sin x f x x x x ' < > eR? MT S BI TP LIN H GIA O HM V TCH PHN MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 114 Gii Khng tn ti. Ta c:( ) ( ) ( ) ( ) ( ) ( )2 2 20 0 00 2 2 sin 2 1 cosx x xf x f f t dt f t f t dt tdt x'' = = > = ( } } } Suy ra:( ) ( ) ( )2 20 2 1 cos 4 f f t t > + > . 4.39. Cho s thc a | | 0;1 e . Xc nh tt c cc hm lin tc khng m trn| | 0;1sao cho cc iu kin sau y c tha mn: a)( )101 f x dx =} b)( )10xf x dx a =}c)( )12 20x f x dx a =}. Gii p dng bt ng thc Bunhiacovski ta c: ( ) ( ) ( ) ( ) ( )2 21 1 1 120 0 0 0. . xf x dx x f x f x dx x f x dx f x dx| | | |= s ||\ . \ .} } } }. M theo gi thit:( ) ( ) ( )21 1 120 0 0. xf x dx x f x dx f x dx| | = |\ .} } }. Do f lin tc trn| | 0;1nn( ) ( ) | | 0,x 0;1 x f x f x = > eSuy ra:( ) | | 0x 0;1 f x = e . iu ny mu thun vi gi thit:( )101 f x dx =}. Vy khng tn ti hm f tho mn bi ton. 4.40. Cho f l hm lin tc trn| ) 0; +v tho mn( ) 0 3 1 xf x < < ( ) 0; x e + . Chng minh rng hm s( ) ( ) ( )330 03x xg x t f t dt tf t dt| |= |\ .} } l hm s ng bin trn( ) 0;+ . Gii Ta c:( ) ( ) ( ) ( ) ( ) ( )2 23 20 09 3x xg x x f x xf x tf t dt xf x x tf t dt (| | | |' = = ( ||\ . \ . } } Li c:( ) ( ) ( )2 22 20 0 0 00 3 1 3 3 0x x x xtf t dt dt x tf t dt x x tf t dt| | | |< < = < > ||\ . \ .} } } } Kt hp vi( ) ( ) 00; xf x x > e + , ta suy ra:( ) ( ) 0x 0; g x ' > e + . Vy( ) g xl hm s ng bin trn( ) 0;+ . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 115 4.41. Cho | ) : 0; f + R kh vi v tho mn( ) ( )( )2 211 1 , f f xx f x' = =+. Chng minh rng tn ti gii hn hu hn( ) limxf x+ v b thua 14t+ . Gii ( )( )| )2 210x 0; f xx f x' = > e ++

f(x) ngbin ( ) ( ) 1 1 x > 1 f x f > = . T ta c:( ) ( )2 11 111 1 arctan 11 4x xxf x f t dt dt ttt' = + < = + < ++} }. Vy tn ti gii hn hu hn( ) limxf x+ v b thua 14t+ . 4.42. Chofl mt hm lin tc trn| ) 0; +tho mn( ) ( )0limxxf x f t dt| |+ |\ .} c gii hn hu hn. Chng minh( ) lim 0xf x= . Gii t( ) ( ) ( ) ( )0xF x f t dt F x f x ' = =}. Khi gi s( ) ( ) ( ) ( ) ( )0lim limxx xf x f t dt F x F x L | |' + = + = |\ .} p dng quy tc Lpitan ta c: ( )( ) ( ) ( )( )( ) ( ) ( )( ) ( ) ( )lim lim lim lim limx x xx xx x x x xxe F x e F x F x e F xF x F x F x Le ee '' +' = = = = + ='Suy ra:( ) ( ) lim lim 0x xf x F x ' = = . 4.43. Hm fxc nh, kh vi trn( ) 0; , + eR. Chng minh rng hm ( ) ( ) f x f x ' +khng gim khi v ch khi( )xf x e'khng gim. Gii t( ) ( ) ( ) h x f x f x ' = +; ( ) ( )xg x f x e' = . Suy ra:( ) ( ) ( )x xe h x e f x '=; ( ) ( )xe g x f x ' = . Khi :( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )00xx x tg x e f x h x e f x h x e f t dt f '' = = = } MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 116 ( ) ( ) ( )00xth x e h t dt f = }. ( ) ( ) ( ) ( ) ( ) ( )00xxh x f x f x e g x f t dt f ' ' = + = + +} =( ) ( ) ( )00xx te g x e g t dt f + +}. ( ) Gi s( ) h xkhng gimKhi vib > a ta c: ( ) ( ) ( ) ( ) ( ) ( )bb a tag b g a e h b e h a e h t dt = }(1) Theo nh l trung bnh ca tch phn tn ti ( ) ( ) ( ) ( )( )1; :b bt t b aa ac a b e h t dt h c e dt h c e e e = = } } (2) Thay (2) vo (1) ta c: ( ) () ( ) ( ) ( ) ( )b a b ag b g a e h b e h a e h c e h c = +( ) ( ) ( ) ( ) ( ) ( )0b ae h b h c e h c h a = + >vib c a > > . Do g(x) khng gim. ( ) :Gi s g(x) khng gim Khi vi b > a ta c: ( ) ( ) ( ) ( ) ( ) ( )bb a tah b h a e g b e g a e g t dt = + } (3) Theo nh l trung bnh ca tch phn tn ti ( ) ( ) ( ) ( )( )1; :b bt t b aa ac a b e g t dt g c e dt g c e e e = = } } (4) Thay (4) vo (3) ta c:( ) ( ) ( ) ( ) ( ) ( )b a b ah b h a e g b e g a e g c e g c = +( ) ( ) ( ) ( ) ( ) ( )0b ae g b g c e g c g a = + > vib c a > > . Do h(x) khng gim. Vy bi ton chng minh xong. 4.44. Chng minh rng nu hm f(x) kh vi v hn ln trnR th hm ( ) ( ) 0 f x fx c nh ngha thm lin tc ti x = 0 cng kh vi v hn ln. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 117 Gii Vi0 x =ta c: ( ) ( ) ( ) ( )( ) ( )( )1 10 0 000xf x ff x f f t dt f ux xdu f ux dux' ' ' = = =} } } V( )10f ux du '} kh vi v hn ln vi mixeR. Vy ( ) ( ) 0 f x fx c nh ngha thm lin tc ti x = 0kh vi v hn ln. 4.45. Tm hm s( ) f xc o hmlin tc trnR sao cho ( ) ( ) ( ) ( )2 2 202011xf x f t f t dt ' = + +}(1). Gii V hm s( ) f xc o hm lin tc trnR nn( )2f xc o hm lin tctrnR. Ly o hm 2 v ca (1), ta c: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )22 22 0 f x f x f x f x f x f x f x f x ' ' ' ' = + = =( )xf x Ce = (2).T (1) suy ra:( ) ( )20 2011 0 2011 f f = = . Cho0 x =, t( ) ( ) 2 0 2011 f C = = . Vy( ) 2011xf x e = . 4.46. Cho a, b >0 . Chng minh: ax0lnbxe e adxx b+ =}. Gii Ta c: ax0 0 0ln ln ln lnb b bbxxt xta a ae e dt adx e dtdx e dxdt t b ax t b+ + + = = = = = =} } } } } } 4.47. Cho hm sfkhng m v lin tc trn| ) 0, +v()0f x dx+< +}. Chng minh rng:()nn01lim xf x dx 0n=}. MT S BI TP V TCH PHN SUY RNG MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 118 Gii t() ( )() ()( )x0F x f x 0F x f t dxF 0 0' = > = =} () () F x 0 F x ' > l hm tng. Bng phng php tch phn tng phn, ta c: () () ( ) () ()n n n0 0 01 1 1xf x dx xd F x F n F x dxn n n= = } } }. Ta bit:() ()nn0limF n f x dx= < +}. By gi, ta s chng minh() ()nn n01lim F x dx limF nn =}. Ta c nh gi sau y: ( ) () ( )nn 1 ni 0 i 101 1 1F i F x dx F in n n= =s s } ( Chng minh ht sc n gin) p dng nh l Kp, ta chng minh c() ()nn n01lim F x dx limF nn =}. Vy()nn01lim xf x dx 0n=}. 4.48. Gi s f , g l cc hm s dng trn| ) a, +v()ag x dx+} phn k. Chng minh rng mt trong cc tch phn sau y phn k. Gii p dng bt ng thc Cauchy, ta c:( )( )1f t 2f t+ > . T suy ra:( )( )( )( )( ) ( )( )( )( )( )x x xa a ag t g tf t g t 2g t f t g t dt dt 2 g t dtf t f t+ > + >} } } Chox + ta c iu cn phi chng minh. 4.49. Tnh 1 txx 0t e dtlim1lnx++ }. Gii Tch phn 1 tat e dt+ } hi t via 0 >c nh bt k , cn x 01lim lnx+| | = + |\ .. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 119 Do 1 tax 0t e dtlim 01lnx++ =}. p dng quy tc LHospital, ta c: a1 t 1 t1 xxx x1x 0 x 0 x 0 x 0t e dt t e dtx elim lim lim lime 11 1xln lnx x+ + + ++ = = = =} }. 4.50. Chng minh rng: cc nh x: lnxf x e x ; 1:xeg xxkh tch trn( | 0,1v 1 10 01lnxxee xdx dxx =} }. Gii Cc nh x f, g lin tc, khng m v( )0ln 0xxx f x e x x+= , ( )011xxeg xx+= suy ra n kh tch trn( | 0,1 . Vi( | 0,1 c e , thc hin tch phn tng phn: 1 1 1 1101ln ln lnx xx xe e dxe xdx e x dx e dxx x xcc c c cc = = + } } } } ( )111 lnxee dxxccc= +}. Suy ra: ( )1 1 1 10 00 01 1ln lim ln lim 1 lnx xx xe ee xdx e dx e dx dxx xcc cc cc+ + | | = = + = |\ .} } } }. 4.51. Tm 22lim1n ndxx x+}. Gii Vi *neN, xt nh x| ) : 2,n + R xc nh bi: ( )211nnxx x = lin tc, khng m v( ) ( )11 x +n nxx+ ~, do nkh tch trn| ) 2, + . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 120 Vi2 n > , nh x 13nxxkh tch trn| ) 2, +v: | )n 21 12,, 03x 1nxxx e + s s Suy ra: ( )122 210 03 3 1 21n n nndx dxx nx x+ + s s = } }. Vy 22lim1n ndxx x+} = 0. 4.52. Tm 2 2limx sxxe e ds++}. Gii Hm s| ) : 0, f + R xc nh bi:( )2sf s e=lin tc khng m V( )20ss f s+, do f kh tch trn| ) 0, + . 0 x > , ta c: 2 2 2 2 22 20 010 02x s x s xt t xtxt s xx xe e ds e ds e dt e dtx+ + + + += < = = s = } } } }. Vy 2 2limx sxxe e ds++} = 0. 4.53. Vi cc gi tr no ca n, tch phn 01n ndxIx+=+} hi t? TmlimnnI ? Gii +R rng0 n s , tch phn nIphn k. + Vi n > 0 , ta c:( )1 1 n +1n nx x +~ . T y, ta thy rng nIhi t vi1 n > . + TmlimnnI ? Ta c: 10 11 1n n ndx dxIx x+= ++ +} } 1 1 110 0 lim 01 1 1n n n nndx dx dxx x n x+ + +< < = =+ +} } }. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 121 Li c: 1 10 011 1nn ndx xdxx x= + +} } 1 10 010 01 1nnn nxdx x dxx n< < = + +} }10lim 01nnnxdxx =+}. 10lim 11nndxx=+}. Vy 1lim 11nndxx+=+}. 4.54. Chng minh rng vi mi0 s > , ta u c: 220s xxsee dx dxx+ '| |< | |\ .} }. Gii Vi mix s > , ta c: 22 sx xsx x e e < >(1) V 2x , x esxx e l hai hm s lin tc, dng trn| ) 0, +nn hai tch phn 2xse dx+} , sxse dx+} u tn ti.T (1) suy ra: 2 220s s xx sxs se ee dx e dx dxs x+ + '| |< = = | |\ .} } }. 4.55. Gi s l mt hm s lin tc trn| ) 0, +v tch phn ( )( )axI a dxx+= } hi t0 a >v hai s dng b, c sao cho b < c. a) Chng minh rng tch phn ( ) ( )0abx cxdxx } hi t v tnh gi tr ca n. b) Chng minh rng tch phn ( ) ( )abx cxdxx +}hi t v gi tr ca n bng ( )( )0lnbacas bdsc s| |+ |\ .}. T suy ra s hi t v gi tr ca tch phn ( ) ( )0bx cxdxx +}. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 122 Gii tt bx = ,d a >ta c: ( ) ( )d bda babx sdx dsx s =} }. Do : ( ) ( )( )a babx sdx ds I bax s + += =} }. Tng t nh trn, ta cng chng minh c : ( )( )acxdx I cax+=}. Vy ( ) ( )( ) ( )( )0a cdbabx cx sdx I ba I ca dsx s = =} } ( pcm). b) t , u=cx u bx = . Vi mi( ) 0, a eeta c : ( ) ( ) ( ) ( ) ( ) ( )a ba ca ba cb c ca bbx cx f s s f s sdx ds ds ds dsx s s s see e e e = = +} } } } }. p dng nh l gi tr trung bnh ca tch phn tn ti( ) , b c e e esao cho ( )() ( )( ) 00ln 0 ln lncbs c c cdss b b beee | |= = |\ .}. Do ( ) ( )( )( )00lna bacabx cx f s bdx dsx c s | |= + |\ .} }. Suy ra : ( ) ( )( ) 00lnbx cx cdxx b +| |= |\ .}. C-MT S BI TP NGH 4.56. Cho f l mt hm s c o hm cp hai trn| | 0,1vf ''b chn v kh tch Riemann. Chng minh rng:()( ) ( )1n2ni 10f 1 f 0 1 2i 1limn f x dx fn 2n 24=' ' | || | = ||\ . \ .}. 4.57. Cho f l mt hm lin tc trn| | 0,1 . Tnh ()21n01nn x0x f x dxlimx e dx}}

MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 123 4.58.Cho f l mt hm lin tc trn| | 0,1 . Tnh () ( )12nn0lim n f x sin 2 x dx t| | |\ .} 4.59. Cho f l mt hm lin tc trn| | 0,1 . Tnh()( )212n01nx 2n0f x sin (2 x)dxlime sin 2 x dxtt}} 4.60. Gi s f v g l hai hm s dng, lin tc trn| | ; a b . Chng minh rng tn ti( ) ; c a b esao cho ( )( )( )( )1c ba cf c g cf x dx g x dx =} }. 4.61. Cho| | ( )10;1 f C e . Chng minh rng tn ti( ) 0;1 cesao cho: ( ) ( ) ( )10102f x dx f f c ' = +}. 4.62. Gi s| | ( ); f C a b e , a > 0 v( ) 0baf x dx =}.Chng minh tn ti ( ) ; c a b esao cho( ) ( )caf x dx cf c =}. 4.63. Chng minh rng phng trnh: 2 4022 xt0t t te 1 ... dt 20111! 2! 4022! | |+ + + + = |\ .} lun c nghim trong trong khong( ) 2011;4022 . 4.64. Chng minh 12012 20110x xdxln x 2013e .2012}=4.65. C tn ti hay khng mt hm s f kh vi lin tc trn| | 0, 2v tho mn ( ) ( ) () | | f 0 f 2 1 ,f x 1x 0, 2 ' = = s e ,()20f x dx 1 s} ? 4.66. Cho hm s() f xlin tc trn| | a, bv c() ( ) f x 0x a, b '' > e . Chng minh rng() ( ) ( ) ( ) ( )baa b 2cf x dx b a f c f c c a, b2+ (' > + e ( } 4.67. Gi s hm s f(x) cng o hm() f x 'lin tc trn| | 0,1 . Chng minh rng:() () ()1 1 10 0 0f x dx f x dx ; f x dx ' s ` )} } }. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 124 4.68. Cho | | ( ) ( ) ( ) ( ){ }2f C 0,1 : f 0 f 1 0 , f 0 a ' = e = = = .Tm() ( )12f0min f x dxe''}. 4.69. Gi s| | ( )10;1 f C ev( ) 0 0 f ' = . Vi( | 0;1 xe , cho( ) x utho mn ( ) ( ) ( )0xf t dt f x x u =}. Tm ( )0limxxxu+. 4.70. Cho hm s f(x) kh vi hai ln trn| ) 0, + . Bit rng() () f x 0,f x 0 ' > >v () ()() ( )| )2f x f x2x 0,f x''s e +'. Chng minh ()() ( )2xf xlim 0f x+'= . 4.71. Chng minh rng: 011 2x 2013 201401x e dx2013 1007tt t22> +}. 4.72. ChoneN. Chng minh rng: 2 2x02e sinnxdx e .ntts} 4.73. Gi s rng:m n 14 + =vim,neN v( ) ( )1nm0I m, n x 1 x dx = }. Hy tm gi tr ln nht, gi tr nh nht ca( ) I m, n . 4.74. Cho f l hm s lin tc trn| | 0,1tho mn: ()1k00 , k = 1, 2 , ..., 2010x f x dx1 , k = 2011= }. Chng minh rng tn ti| | c 0,1 esao cho:( )2012f c 1006.2 > . 4.75. Cho | || | ( )k k a ,bf C; x a, b; > 0 k = 1 , 2 ,..., 2011 e e . Chng minh rng tn ti| | c a, b esao cho:()kx2011kk 0cf x dx 0 ==}. 4.76.Cho( )2 411, xnxx n== e+R. Hy tnh( )0F t dt+}. 4.77. Tm m tch phn sau hi t: ( )11 I x m x x x dx+= + }. 4.78. Cho| ) : 0, f + R l hm kh vi v tha mn hai iu kin: a)( ) ( ) ( ) ( )3 2 2, x > 0 f x f x f x ' = b)( ) , x 0xf x es > .Hy biu din ( )0 , nnnu x f x dx+= e}N qua 0uv chng minh dy 3lim .!nnxun| | < + |\ .. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 125 CHNG 5 A THC VI MT S YU T GII TCH A. TM TTL THUYT I. GII HN V LIN TC Cho| | f x eR ,()n n 1o 1 n 1 nf x a x a x ... a x a= + + + +1. Gii hn a)()0x0khi a 0limf xkhi a 0++ >= 0 , n = 2k , khay a < 0 , n = 2k+1 , klimf xkhi a > 0 , n = 2k+1 , khay a eR hoc() f x 0 ' s x eR th f(x) = 0 c duy nht nghim. (ii) Nu phngtrnh () f x 0 ' =c hai nghim phn bit th th hm s c hai cc tr. +Vi yC.yCT > 0 th phng trnh f(x) = 0 ch c 1 nghim +Vi yC.yCT = 0 th phng trnh f(x) = 0c 2 nghim( 1 nghim n, 1 nghim kp) +Vi yC.yCT < 0 th phng trnh f(x) = 0 c nghim phn bit MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 126 e) Cc nh l gi tr trung bnh, khai trin Taylor, Quy tc LHospital u c th s dng c i vi a thc (bn c xem li vn ny trong l thuyt v O HM). III. NGUYN HM Cho| | f x eR ,()n n 1o 1 n 1 nf x a x a x ... a x a= + + + +c nguyn hm l:() ()n 1 n 0 1na aF x f x dx x x ... a x Cn 1 n+= = + + + ++} ( C l hng s tu ). B. BI TP P DNG 5.1. Liu c tn ti hay khng hai a thc f(x) v g(x) tha: ()()f x 1 1 11 ...g x 2 3 n= + + + + , *n eN . Gii D thy: n1 1 1lim 1 ...2 3 n| |+ + + + = + |\ .. Gi s tn ti hai a thc f(x) v g(x) tho yu cu bi ton th ()()xf xlimg x= +. Suy ra:degf degg > . Khi ()()0x0f xlim axg xb

=

(*) (00a , bln lt l h s ca n x bc cao nht ca f(x) , g(x) ) Chng minh n1 1 1 1lim 1 ... 0n 2 3 n| |+ + + + = |\ . (**)hon ton khng kh khn. (*) v (**) mu thun nhau.Vy khng tn ti hai a thc f(x) v g(x) tho mn yu cu ca bi ton. 5.2. Vi f(x) l a thc bc n v cc sa b > > > . Chng minh rng tt c cc nghim thc ca a thc f(x) u thuc khong(a, b). Gii p dng khai trin Taylor, ta c: () ( )( )( )( )( )( )( )( )n2 nf a f a f af x f a x a x a ... x a1! 2! n!' ''= + + + + MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 127 =( )( )( )( )( )( )( )( )( )n2 nf a f a 1 f af a a x a x ... a x1! 2! n!' '' + + + + Nux a sth() f x 0 < . Suy ra() f xkhng c nghimx a s . () ( )( )( )( )( )()( )( )n2 nf b f b f bf x f b x b x b ... x b1! 2! n!' ''= + + + + Nux b >th() f x 0 > . Suy ra() f xkhng c nghimx b > . Vy ta c iu phi chng minh. 5.3. Cho 0 1 2 2011a ,a , a ,..., a eR v tho mn iu kin sau y: 2 20111 2 2011 2 20110 0 1a a a a 2 a 2a ... a a ... 02 3 2012 3 2012+ + + + = + + + + = . Chng minh phng trnh: 20101 2 2011a 2a x ... 2011a x 0 + + + =c t nht mt nghim thuc khong( ) 0, 2 . Gii Xt a thc:()2 3 20120 1 2 20111 1 1f x a x a x a x ... a x2 3 2012= + + + +R rng f(x) lin tc trnR. Da vo gi thit bi ton ta d dng chng minh c:( ) ( ) ( ) f 0 f 1 f 2 0 = = = . p dng nh l Rolle tn ti( ) ( ) a 0,1, b 1, 2 e esao cho( ) ( ) f a f b 0 ' ' = = . Li p dng nh l rolle : tn ti( ) ( ) c a, b 0, 2 e csao cho( ) f c 0 '' = . M()20101 2 2011f x a 2a x ... 2011a x '' = + + + . Vy ta c iu phi chng minh. 5.4. Cho a thc f(x) c 3 nghima b c < | | x 0,1 e . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 129 Suy ra:( ) f 0 8 ' = . Vy gi tr nh nht ca m bng 8. 5.7. Tnh ( )| | ( )limxP xP x ( , y( ) P xl a thc vi h s dng. Gii V P l a thc vi h s dng , vi x > 1 ta c: ( )( )( )| | ( )( )( )11P x P x P xP x P x P x ( s s. V ( )( )( )( )1lim lim 11x xP x P xP x P x = = nn ( )| | ( )lim 1xP xP x ( = . 5.8. Cho a thc()6 5 4 3 2Q x 2x 4x 3x 5x 3x 4x 2 = + + + + + + . t () () ()2 30 0 0x x xI dx , J= dx, K= dxQ x Q x Q x+ + += } } }. Chng minh rng:I K J = > . Gii + a thc Q(x) c vit li nh sau: () ( ) ( ) ( )6 5 4 2 3Q x 2 x 1 4 x x 3 x x 5x = + + + + + + . + R rng cc tch phn trn hi t. + t 1xt= , ta c: 206 5 4 2 31 1. dtt tI1 1 1 1 1 12 1 4 3 5t t t t t t+=| | | | | |+ + + + + + |||\ . \ . \ .} + Ta thu gn biu thc di du tch phn I s thu cI K = . + Li c:( )()( )()23 20 0x x 1 x 2x x2 I J I K 2J dx dx 0 I JQ x Q x+ + + = + = = > >} }. VyI K J = > . 5.9. Cho()4 3 2f x x 2010x 2011x 2012x 2013 = + + + +v bit rng() f x 0 >vi mix eR. Chng minh rng: () () () () ()( )()4g x f x f x f x f x f x 0x ' '' ''' = + + + + > eR. Gii V g(x) l a thc bc 4 nn()xlimg x= +. Do tn ti ( ) ()0 0xx : g x ming xee =RR . T y suy ra:( )0g x 0 ' = . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 130 V() () () g x g x f x ' = nn( ) ( ) ( )0 0 0g x 0 g x f x 0 ' = = > . M() ( )0g x g x >nn() g x 0x > eR. 5.10.Cho P(x) l mt a thc c bc n > 3 v 1 2 3 1...n nx x x x x< < < < ,x eR. Gii + Nu n = 1 th khng nh trn lun ng. + Nu2 n > , gi 1,...,n l cc nghim ca a thc P(x).Khi vi , k = 1,...,nkx =th khng nh trn hin nhin ng. By gi, gi s , k =1,...,nkx =ta c : ( )( )()( ) ( )( )1 1P x1 2 ;nk k l ni k lP xP x x P x x x = s < s' ''= = . Do : ( )( )( )( )( )( )( )( )2 21 11 21 1nk k l nk k lP x P xn n n nP x P x x x x = s < s| | ' '' | | = || \ . \ . =( )( )( ) ( )( )21 1 11 1 21 2nk k l n k l nk k l k ln nx x x x x = s < s s < s (| || | ( + || | (\ .\ . = ( )( )( )( ) ( )( )21 1 11 1 22 1nk k l n k l nk l k lknn nx x x xx = s < s s < s| |+ | | \ . = ( )( )( )221 1 11 2 1 10nk k l n k l nk l k lknx x x xx = s < s s < s| | = > | \ . . Vy( )( )( )( )( )( ) ( ) ( ) ( )221 0 1 .P x P xn n n P x nP x P xP x P x| | ' ''' '' > > |\ . 5.17. Cho P(x) l a thc bc n vi h s thc c n nghim thc phn bit khc 0. Chng minh rng cc nghim ca a thc( ) ( ) ( )23 x P x xP x P x '' ' + +l thc v phn bit. Gii MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 134 t( ) ( ) Q x xP x = . V cc nghim ca P(x) l thc phn bit khc 0 nn cc nghim cu Q(x) cng l thc phn bit. Theo nh l Rolle suy ra cc nghim ca( ) Q x 'lthc phn bit. t( ) ( ) R x xQ x ' =, suy ra cc nghim ca R(x) l nghim thc phn bit. Do cc nghim ca a thc( ) ( ) ( ) ( )23 R x x P x xP x P x ' '' ' = + +l thc phn bit. 5.18. Cho P(x) l a thc vi h s thc c n nghim thc phn bit ln hn 1. Xt a thc ( ) ( ) ( ) ( ) ( ) ( )2 2 21 Q x x P x P x x P x P x ' '( = + + + . Chng minh rng a thc Q(x) c2 1 n nghim thc phn bit. Gii Ta c :( ) ( ) ( ) ( ) ( ) Q x P x xP x xP x P x ' ' = + +(( . Nhn xt : ( ) ( ) ( )( ) ( ) ( ) ( )2 22 2x xP x xP x e e P xxP x P x xP x'| | ' + = | |\ .' ' + = Gi cc nghim ca P(x) l : 1 21 ...nx x x < < < < . Theo nh l Rolle, ( ) ( ) P x xP x ' +c1 n nghim , k = 1,2,...,n-1kytha: 1 1 2 11 ...n nx y x y x< < < < < suy ra :( ) 0 P q =nhng li c : 1 k kx q x +< < . iu ny mu thun.5.19. Tm tt c cc a thc P(x) h s thc tha mn iu kin : ( ) ( )2011 xP e P x =vi mixeR. Gii T gi thit suy ra : ( )( )20110xP e P x x = > . t 201110 n1 , u, n 1nuu e= = > . MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 135 Khi ( )nul dy tng v( ) ( )1nn nP u P u = eN. Suy ra( ) P x C = . 5.20. Cho a thc lng gic( ) ( )k0oskx+b sinnkkP x a c kx== tha mn iu kin ( ) 1x P xs eR. Chng minh rng :( ) P x n x ' s eR. Gii ChoaeR bt k. D thy :( )( ) ( )1sin2nkkP a x P a xQ x c kx=+ = = Suy ra :( )( ) ( )2P a x P a xQ x' ' + ' =v( ) ( ) 0 Q P a ' ' = . Ta chng minh( ) 0 Q n ' s . Tht vy( )( ) ( )1.2P a x P a xQ x+ + s sNhn xt : ( )sinxQ xn s{ } ..., 2 , ,0, , 2 ,... x t t t t = ( nhn xt ny ti dnh cho bn c t kim tra). Hn na( ) 0 0 Q =v ( ) ( ) 0.0 sinxQ x Q xnxs. Khi0 x ta thu c( ) 0 Q n ' shay( ) P a n s . V a c ly bt k nn ta suy ra :( )x P x n s eR. C-MT S BI TP NGH 5.21. Cho P l mt a thc bc n tha mn:( )100, k = 1,2,...,nkx P x dx =}. Chng minh rng:( ) ( ) ( )21 1220 01 P x dx n P x dx| |= + |\ .} }. 5.22. Tm mt a thc bc nh nht, t gi tr cc tiu l 2 ti im 13 x =v t gi tr cc i l 6 ti im 21 x = . 5.23. Cho a thc( ) ( )( )2 3 21 ... 1 ...2! 3! ! 2 !k nk x x x xP x xk n= + + + + +( ) neNc gi tr khng m vi mi x. 5.24. Cho a thc( ) P xvideg P n =v( ) 0P x x > eR. Chng minh rng : MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 136 ( )( )00nkkP x=>. 5.25. Cho a thc:( )1 21 2 1...n n nn nP x x a x a x a x a = + + + + +tha mn iu kin:( ) ( ) ( )( ) ( )1 21x n n P x x x x x P x '' = eR v c cc nghim thc trn | |1 2, x xth cc nghim y c th xp thnh cp i xng nhau qua 1 202x xx+= . 5.26. Gi s Q(x) l a thc vi h s thc v c bc ln hn v bng 1. Chng minh rng ch c mt s hu hn cc gi tr ca m sao cho ( ) ( )0 0sin x cos 0m mQ x dx Q x xdx = =} }. 5.27. Gi s Q(x) l a thc khng c nghim thc. Chng minh rng a thc( )( )( )( )( )( )( )( )( )4 6 2... ....2! 4! 6! 2 !nQ x Q x Q x Q xQ xn''+ + + + + +cng khng c nghim thc. 5.28. Cho a thc 10 1 1 0( ) ... , a 0n nn nP x a x a x a x a= + + + + =c n nghim thc phn bit. Chng minh rng : a)( ) ( ) 2011 2012 0 P x P x ' =c n nghim thc phn bit. b)21 0 212na a an> . 5.29. Cho a thc P(x) bc n c n nghim thc phn bit. Chng minh tp hp nghim ca bt phng trnh : ( ) ( )( )20110P x P xP x' >l mt s khong c tng di bng 2011n. 5.30. Cho a thc P(x) bc n > 1 c n nghim thc phn bit , k = 1,2,...,nk . Chng minh rng : ( ) ( ) ( )1 21 1 1... 0nP P P + + + =' ' '. MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 137 CHNG 6BI TP NNG CAOV NHNG GI V PHNG PHP GII 6.1. Cho hm s: f R R tha mn iu kin:( ) ( ) 19 19 f x f x + s +v ( ) ( ) 94 94 f x f x + > +vi mi x. Chng minh rng:( ) ( ) 1 f x f x + =vi mi xeR. Gi Bc 1: Chng minh bng quy np vi mineN ( ) ( ) 19 19 f x n f x n + s + , ( ) ( ) 94 94 f x n f x n + > +( ) ( ) 19 19 f x n f x n > ,( ) ( ) 94 94 f x n f x n s . Bc 2: Chng minh( ) ( ) ( ) 1 1 1 f x f x f x + s + s + . 6.2. Cho f lin tc trn on| | ; a b , kh vi trong khong( ) ; a bv ( ) ( ) 0 f a f b = = . Chng minh rng tn ti( ) ; c a b esao cho:( ) ( )2011f c f c ' = . Gi p dng nh l Rolle cho hm s:( )( )( )2010xaf t dtg x e f x}= . 6.3.C hay khng mt hm s: f R R tha mn: ( ) sin sin 2 f x y x y + + + < vi x, y eR ? Gi Bc 1: Chn cc gi tr thch hp ca x, y thay vo iu kin bi ton thu c:( ) 2 2 ft + < ; ( ) 2 2 ft < . Bc 2: S dng bt ng thc tam gic dn n mt iu mu thun vi gi thit. Khi ta kt lun c khng tn ti hm s no tha mn yu cu bi ton. 6.4.Tm hm s( ) f xc o hmlin tc trnR sao cho ( ) ( ) ( ) ( )2 2 202011xf x f t f t dt ' = + +} Gi Ly o hm hai v. 6.5. Cho: f R R sao cho( ) ( ) a b f a f b a b < = . Chng minh rng nu( ) ( ) ( )0 0 f f f =th( ) 0 0 f = . Gi Bc 1: Vit li iu kin i vi hm f(x) :( ) ( ) f a f b a b s MATHVN.COMwww.MATHVN.com BI TP GII TCH DNH CHO OLYMPIC TON VN PH QUC- GV. TRNG I HC QUNG NAM 138 Bc 2: t( ) ( ) 0 ,y = f x f x = . Khi () 0. f y =p dng bt ng thc trn lin tip ta thu c0 x y = = . Suy ra( ) 0 0 f = . 6.6. Cho f xc nh trn| | 0;1tho mn:( ) ( ) 0 1 0 f f = =v ( ) () | | x, y 0,12x yf f x f y+| |s + e |\ .. Chng minh rng: phng trnh( ) 0 f x =c v s nghim tr