Phu Dao Dai So 10 CA Nam

8/9/2019 Phu Dao Dai So 10 CA Nam

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PHN 1 HM S BC NHT y ax b I. Kin thc c bn:

1. Hm s 0 y ax b a :

- Tp xc nh D R .- Hm s y ax b ng bin trn 0R a - Hm s y ax b nghch bin trn 0R a

- th l ng thng qua 0; , ;0b

A b Ba

.

2. Hm s hng b :- Tp xc nh D R .

- th hm s b l ng thng song song vi trc honh Ox v i qua 0;A b .

3. Hm s x :

- Tp xc nh D R .- Hm s x l hm s chn.

- Hm s ng bin trn 0; .

- Hm s nghch bin trn ;0 .

4. nh l: :d y ax b v ' : ' 'd y a x b

- d song song 'd 'a a v 'b b .

- d trng 'd 'a a v 'b b .

- d ct 'd 'a a .

Bi tp v d:

1) V th ca cc hm s sau trn cng mt h trc ta : 2x ; 2 2y x ; 3x ; 2y Hm s 2y x Hm s 2 2y x Hm s 3x

Cho 0 0x y , 0;0O cho 0 2x y , 0; 2B cho 0 3x y , 0;3D Cho 1 2x y , 1;2A cho 1 0x y , 1;0C cho 1 2x y , 1;2A

Hm s 2y l ng thng song song vi trc honh Ox v i qua im 0;2E

(Hc sinh t v hnh)2) Tm a,b th hm s ax b i qua hai im 2;1A v 1;3B .Gii: V th hm s y ax b i qua hai im 2;1A v 1;4B nn ta c h phng trnh

2 1

4

a b

a b

Gii h ta c 1a v 3b . Vy hm s cn tm l 3x .

3) Tm ta giao im ca th hai hm s bc nht: tm ta giao im (nu c) ca thhai hm s bc nht sau y 2 1x v 3 2x .Gii: Ta giao im l nghim ca h

2 1 2 1 3 2 1

3 2 3 2 1

x x x x

x y x y

.

Vy giao im cn tm l im 1;1M

4) Tm a,b ng thng ax b i qua 1;1M v song song vi ng thng 3 2y x Gii: V ng thng y ax b song song vi ng thng 3 2y x nn ta c 3a .


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V ax b i qua 1;1M nn ta c 1 1.a b , th 3a ta tm c 4b Vy ng thng cn tm l 3 4y x .

5) V th hm s cho bi nhiu cng thc:V th hm s

1, khi 1

2 , khi 1

x x y f x

x x

Vi 1x ta c 1x Vi 1x ta c 2 x Cho 1 2x y , 1;2A cho 0 2x y , 0;2C

Cho 2 3x y , 2;3B cho 1 3x y , 1;3D

BI TP

1. V th ca cc hm s sau trn cng mt h trc ta : 2 ; 2 ; 2 3 ; 2y x y x y x y .2. V th ca cc hm s sau:

a) 1, khi 02 , khi 0

x xy

x x

b) 3 1, khi 1

1, khi 1x x

yx x

c) 2 4, khi 24 2 , khi 2

x xy

x x

d)

2, khi 1

2 1, khi 1

x xy

x x

e) 1x f) 2 3y x g) 1y x h) 1 2y x

3. Tm m cc hm s:a) 1 3 y m x ng bin trn R . b) 2 3 6m x nghch bin trn R .

c) 1 3 2 y m x x m tng trn R . d) 2 3 2m x x m gim trn R .

4. Tm a,b th hm s y ax b :a) i qua hai im 1; 3A v 2;3B . c) i qua im 2; 1M v song song vi

3x

b) i qua gc ta v 2;1A . d) i qua gc ta v song song vi

2 2009y x

5. Tm m :a) th hm s 3 5x ct th hm s 2 5m x .


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b) th hm s 2 2y x song song vi th hm s 2 1 2m x m .

c) th hm s 2x trng vi th hm s 2 2y m x m .

6. Tm ta giao im nu c ca th hai ham s:a) 3 1y x v 1x b) 3 1x v 1x c) 5 6y x v 6x

7. Tm m th ca ba hm s sau ng quy (cng i qua mt im):a) 2y x v 3y x v 1 y mx

b) 1x v 3 x v 2 3 2y m x m

c) 2y x v 3 y x m v 2 5m x

8. Cho hm s 1 2 y m x a) Chng minh rng th hm s trn lun i qua mt im c nh vi mi m .

b) Tm 0m th hm s 1 2 y m x ct ,Ox Oy ti hai im ,A B sao cho OAB cn ti O.

PHN 2

Hm s bc hai - mt s dng ton lin quan

Dng 1. Kho st s bin thin v v th

Bi 1. Kho st s bin thin v v th cc hm s sau:

a)y= x2- 6x+ 3 b)y= x2- 4x+ 3 c)y= -x2 + 5x- 4

d) y= 3x2+ 7x+ 2 e) y= -x2- 2x+ 4

Bi 2. Kho st s bin thin v v th cc hm s sau:

a) 2 y x 4x 3 b) 2 y x 4 x 3 c) 2x 4 x 3

d) 2y x 4 x 3 e) 2 y x 4 x 3

Bi 3. Tm gi tr ln nht, nh nht ca hm s:

a) y = x2 -5x + 7 trn on [-2;5] b) y = -2x2 + x -3 trn on [1;3]

c) y = -3x2 - x + 4 trn on [-2;3] d) y = x2 + 3x -5 trn on [-4; -1]

Bi 4. Tm m cc bt phng trnh sau ng vi mi gi tr ca m:

a) x2 - 3x + 1 > m b) -x2 +2x - 1 > 4m c) 22x x 1 2m 1


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d) 23x x 3 3m e) x 1 x 2 x 3 x 4 m f) 2 2 x 2x 1 m m

g) x 3 x 5 x 2 x 4 3m 1

Dng 2. Lp ph-ng trnh ca parabol khi bit cc yu t ca n

Bi 5. Xc nh phng trnh cc parabol:

a) y= x2+ ax+ b i qua S(0; 1)

b) y= ax2+ x+ b i qua S(1; -1)c) y= ax2+ bx- 2 i qua S(1; 2)

d) y= ax2+ bx+ c i qua ba im A(1; -1), B(2; 3), C(-1; -3)

e) y= ax2+ bx+ c ct trc honh ti x1= 2v x2= 3, ct trc tung ti: y= 6

f) y= ax2+ bx+ c i qua hai im m(2; -7), N(-5; 0) v c trc i xng x= -2

g) y= ax2+ bx+ c t cc tiu bng6 ti x= -3 v qua im E(1; -2)

h) y= ax2+ bx+ c t cc i bng 7 ti x= 2 v qua im F(-1; -2)

i) y= ax2+ bx+ c qua S(-2; 4) v A(0; 6)

Bi 6. Tm parabol y=ax2+ bx+ 2 bit rng parabol :

a) i qua hai im A(1; 5) v B(-2; 8) b)Ct trc honh ti x1= 1 v x2= 2

c) i qua im C(1; -1) v c trc i xng x= 2 d)t cc tiu bng 3/2 ti x= -1

e) t cc i bng 3 ti x= 1

Bi 7. Tm parabol y= ax2+ 6x+ c bit rng parabol

a) i qua hai im A(1; -2) v B(-1; -10) b)Ct trc honh ti x1= -2 v x2= -4

c) i qua im C(2; 5) v c trc i xng x= 1 d)t cc tiu bng -1 ti x= -1

e) t cc i bng 2 ti x= 3Bi 8. Lp phng trnh ca (P) y = ax2 + bx + c bit (P) i qua A(-1;0) v tip xc vi ng

thng (d) y = 5x +1 ti im M c honh x = 1

Dng 3. S t-ng giao ca parabol v -ng thng

Bi 9. Tm to giao im ca cc hm s sau:

a) y= x- 1 v y= x2- 2x- 1 b) y=-x+ 3 v y= -x2- 4x +1

c) y= 2x- 5 v y=x2- 4x+ 4 d) y= 2x+ 1 v y=x2- x- 2

e) y= 3x- 2 v y= -x2- 3x+ 1 f) y= -4

1x+ 3 v y=

2

1x2+ 4x+ 3

Bi 10. Tm to giao im ca cc hm s sau:

a) y= 2x2+3x+ 2 v y= -x2+ x- 1 b) y= 4x2- 8x+ 4 v y= -2x2+ 4x- 2

c) y= 3x2+ 10x+ 7 v y= -4x2+ 3x+ 1 d)y= x2- 6x+ 8 v y= 4x2- 5x+ 3

e)y= -x2+ 6x- 9 v y= -x2+ 2x+ 3 f) y= x2- 4 v y= -x2+ 4

Bi 11 Bin lun s giao im ca ng thng (d) vi parabol (P)


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a) (d): y= mx- 1 v (P): y= x2- 3x+ 2

b) (d): y= x- 3m+ 2 v (P): y= x2- x

c) (d): y= (m- 1)x+ 3 v (P): y= -x2+ 2x+ 3

d) (d): y= 5x+ 2m+ 5 v (P): y= 5x2+ 3x- 7

Bi 12. Cho h (Pm) y = mx2 + 2(m-1)x + 3(m-1) vi m0. Hy vit phng trnh ca parabol

thuc h (Pm) tip xc vi Ox.

Bi 13Cho h (Pm) y = x2 + (2m+1)x + m21. Chng minh rng vi mi m th (Pm) lun ct

ng thng y = x ti hai im phn bit v khong cch gia hai im bng hng s.

Dng 4. Ph-ng trnh tip tuyn ca Parabol

Bi 14. Vit phng trnh tip tuyn ca (P) y = x2 - 2x +4 bit tip tuyn:

a) Tip im l M(2;4) b) Tip tuyn song song vi ng thng (d1) y = -2x + 1

c) Tip tuyn i qua im A(1:2) d) Tip tuyn vung gc vi (d2) y = 3x + 2

Bi 15. Vit phng trnh tip tuyn ca (P) y = -2x2 + 3x -1 bit tip tuyn:

a) Tip im l M(-1;3) b) Tip tuyn song song vi ng thng (d1) y = 3x -2c) Tip tuyn i qua im A(-3:2) d) Tip tuyn vung gc vi (d2) y = -3x -1

Dng 5. im c bit ca Parabol

Bi 16. Tm im c nh ca (Pm): y = mx2 + 2(m-2)x - 3m +1.

Bi 17. Tm im c nh ca (Pm): y = (m+1)x2 - 3(m+1)x - 2m -1

Bi 18. Tm im c nh ca (Pm): y = (m2 - 1)x2 - 3(m+1)x - m2 -3m + 2

Dng 6. Qu tch im

Bi 19. Tm qu tch nh ca (Pm) y = x2 - mx + m

Bi 20. Tm qu tch nh ca (Pm) y = x2 - (2m+1)x + m-1

Bi 21. Cho (P) y = x2

a) Tm qu tch cc im m t c th k c ng hai tip tuyn ti (P).

b) Tm qu tch tt c cc im m t ta c th k c hai tip tuyn ti (P) v hai tip tuyn

vung gc vi nhau.

Dng 7. Khong cch gia hai im lin quan n parabol

Bi 22. Cho (P)2x

y

4

v im M(0;-2). Gi (d) l ng thng qua M c h s gc k

a) Chng t vi mi m, (d) lun ct (P) ti hai im phn bit A v B.

b) Tm k AB ngn nht.

Bi 23. Cho (P) y = x2, ly hai im thuc (P) l A(-1;1) v B(3;9) v M l mt im thuc cung

AB. Tm to ca M din tch tam gic AMB l ln nht.

Bi 24.Cho hm s y = x2 +(2m+1)x + m2 - 1 c th (P).


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a) Chng minh rng vi mi m, th (P) lun ct ng thng y = x ti hai im phn bit v

khong cch gia hai im ny khng i.

b) Chng minh rng vi mi m, (P) lun tip xc vi mt ng thng c nh. Tm phng trnh

ng thng .

Bi 25. Cho (P) 2 y 2x x 3 . Gi A v B l hai im di ng trn (P) sao cho AB=4. Tm qu

tch trung im I ca AB.Dng 8. ng dng ca th trong gii ph-ng trnh, bpt

Bi 26. Bin lun theo m s nghim ca phng trnh:

a) x2 + 2x + 1 = m b) x2 -3x + 2 + 5m = 0 c) - x2 + 5x -6 - 3m = 0

Bi 27. Bin lun theo m s nghim ca phng trnh:

a) 2 x 5x 6 3m 1 b) 2 x 4 x 3 2m 3 c) 22x x 4m 3 0

Bi 28. Tm m phng trnh sau c nghim duy nht: 2

2 2 x 2x 4 x 2x 5 m

Bi 29. Tm m phng trnh sau c 4 nghim phn bit: 2 x x 2 4m 3

Bi 30. Tm m phng trnh sau c 3 nghim phn bit: 2 x x 2 5 2m

Bi 31. Tm gi tr ln nht, nh nht ca ( ) 4 3 2 f x x 4x x 10x 3 trn on [-1;4]

Bi 32. Cho x, y, z thay i tho mn x2 + y2 + z2 = 1. Tm gi tr ln nht v nh nht ca P= x +

y + z + xy+ yz + zx

Bi 33. Tm m bt ng thc 2 2 x 2x 1 m 0 tho mn vi mi x thuc on [1;2].

PHN III


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Ph-ng trnh bc hai & h thc VitBi tp 1 : nh gi tr ca tham s m ph-ng trnh

2 ( 1) 5 20 0 x m m x m C mt nghim x = 5 . Tm nghim kia.

Bi tp 2 : Cho ph-ng trnh2 3 0 x mx (1)

a) nh m ph-ng trnh c hai nghim phn bit.

b) Vi gi tr no ca m th ph-ng trnh (1) c mt nghim bng 1? Tm nghim kia.Bi tp 3 : Cho ph-ng trnh

2 8 5 0 x x m (1)a) nh m ph-ng trnh c hai nghim phn bit.b) Vi gi tr no ca m th ph-ng trnh (1) c mt nghim gp 3 ln nghim kia? Tm ccnghim ca ph-ng trnh trong tr-ng hp ny.Bi tp 4 : Cho ph-ng trnh

2( 4) 2 2 0m x mx m (1)

a) m = ? th (1) c nghim l x = 2 .b) m = ? th (1) c nghim kp.Bi tp 5 : Cho ph-ng trnh

2 2( 1) 4 0 x m x m (1)a) Chng minh (1) c hai nghim vi mi m.b) m =? th (1) c hai nghim tri du .c) Gi s 1 2,x x l nghim ca ph-ng trnh (1) CMR : M = 2 1 1 21 1 x x x x khng ph

thuc m.Bi tp 6 : Cho ph-ng trnh

2 2( 1) 3 0 x m x m (1)a) Chng minh (1) c nghim vi mi m.b) t M = 2 21 2x x ( 1 2,x x l nghim ca ph-ng trnh (1)). Tm min M.

Bi tp 7: Cho 3 ph-ng trnh2

2

2

1 0(1);

1 0(2);

1 0(3).

x ax b

x bx c

x cx a

Chng minh rng trong 3 ph-ng trnh t nht mt ph-ng trnh c nghim.Bi tp 8: Cho ph-ng trnh

2 2( 1) 2 0 x a x a a (1)a) Chng minh (1) c hai nghim tri duvi mi a.b) 1 2,x x l nghim ca ph-ng trnh (1) . Tm min B =

2 21 2x x .

Bi tp 9: Cho ph-ng trnh

2 2( 1) 2 5 0 x a x a (1)a) Chng minh (1) c hai nghim vi mi ab) a = ? th (1) c hai nghim 1 2,x x tho mn 1 21x x .

c) a = ? th (1) c hai nghim 1 2,x x tho mn2 21 2x x = 6.

Bi tp 10: Cho ph-ng trnh22 (2 1) 1 0 x m x m (1)


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a) m = ? th (1) c hai nghim 1 2,x x tho mn 1 23 4 11x x .b) Chng minh (1) khng c hai nghim d-ng.c) Tm h thc lin h gia 1 2,x x khng ph thuc m.Gi : Gi s (1) c hai nghim d-ng > v lBi tp 11: Cho hai ph-ng trnh

2

2

(2 ) 3 0(1)

( 3 ) 6 0(2)

x m n x m

x m n x

Tm m v n (1) v (2) t-ng -ng .Bi tp 12: Cho ph-ng trnh

2 0( 0)ax bx c a (1)iu kin cn v ph-ng trnh (1) c nghim ny gp k ln nghim kia l

2 2( 1) 0( 0)kb k ac k Bi tp 13: Cho ph-ng trnh

2 2( 4) 7 0mx m x m (1)

a) Tm m ph-ng trnh c hai nghim phn bit 1 2,x x .

b) Tm m ph-ng trnh c hai nghim 1 2,x x tho mn 1 22 0x x .c) Tm mt h thc gia 1 2,x x c lp vi m.Bi tp 14: Cho ph-ng trnh

2 2(2 3) 3 2 0 x m x m m (1)a) Chng minh rng ph-ng trnh c nghim vi mi m.b) Tm m ph-ong trnh c hai nghim i nhau .c) Tm mt h thc gia 1 2,x x c lp vi m.Bi tp 15: Cho ph-ng trnh

2( 2) 2( 4) ( 4)( 2) 0m x m x m m (1)a) Vi gi tr no ca m th ph-ng trnh (1) c nghim kp.b) Gi s ph-ng trnh c hai nghim 1 2,x x . Tm mt h thc gia 1 2,x x c lp vi m.

c) Tnh theo m biu thc 1 2

1 11 1A x x ;

d) Tm m A = 2.

Bi tp 16: Cho ph-ng trnh2 4 0 x mx (1)

a) CMR ph-ng trnh c hai nghim phn bit vi mi .

b) Tm gi tr ln nht ca biu thc 1 22 21 2

2( ) 7x xA

x x

.

c) Tm cc gi tr ca m sao cho hai nghim ca ph-ng trnh u l nghim nguyn.Bi tp 17: Vi gi tr no ca k th ph-ng trnh 2 7 0 x kx c hai nghim hn km nhau

mt n v.

Bi tp 18: Cho ph-ng trnh2 ( 2) 1 0 x m x m (1)

a) Tm m ph-ng trnh c hai nghim tri du.b) Tm m ph-ng trnh c hai nghim d-ng phn bit.


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c) Tm m ph-ng trnh c nghim m.Bi tp 19: Cho ph-ng trnh

2 ( 1) 0 x m x m (1)a) CMR ph-ng rnh (1) lun c nghim phn bit vi mi mb) Gi 1 2,x x l hai nghim ca ph-ng trnh . Tnh

2 21 2x x theo m.

c) Tm m ph-ng trnh (1) c hai nghim 1 2,x x tho mn2 21 2x x = 5.

Bi tp 20: Cho ph-ng trnh

2 2(2 1) 3 0 x m x m m (1)a) Gii ph-ng trnh (1) vi m = 3.b) Tm m ph-ng trnh c hai nghim v tch hai nghim bng 4. Tm hai nghim .Bi tp 21: Cho ph-ng trnh

2 12 0 x x m (1)Tm m ph-ng trnh c hai nghim 1 2,x x to mn

22 1x x .

Bi tp 22: Cho ph-ng trnh2( 2) 2 1 0m x mx (1)

a) Gii ph-ng trnh vi m = 2.b) Tm m ph-ng trnh c nghim.

c) Tm m ph-ng trnh c hai nghim phn bit .d) Tm m ph-ng trnh c hai nghim 1 2,x x tho mn 1 21 2 1 2 1x x .

Bi tp 23: Cho ph-ng trnh2 2( 1) 3 0 x m x m (1)

a) Gii ph-ng trnh vi m = 5.b) CMR ph-ng trnh (1) lun c hai nghim phn bit vi mi m.

c) Tnh A =3 31 2

1 1x x

theo m.

d) Tm m ph-ng trnh (1) c hai nghim i nhau.Bi tp 24: Cho ph-ng trnh

2

( 2) 2 4 0m x mx m (1)a) Tm m ph-ng trnh (1) l ph-ng trnh bc hai.

b) Gii ph-ng trnh khi m =32

.

c) Tm m ph-ng trnh (1) c hai nghim phn bit khng m.Bi tp 25: Cho ph-ng trnh

2 0 x px q (1)

a) Gii ph-ng trnh khi p = 3 3 ; q = 3 3 .b) Tm p , q ph-ng trnh (1) c hai nghim : 1 22, 1x x

c) CMR : nu (1) c hai nghim d-ng 1 2,x x th ph-ng trnh2 1 0qx px c hai nghim

d-ng 3 4,x x

d) Lp ph-ng trnh bc hai c hai nghim l 1 23 3 x v a x ; 21

1

xv

22

1

x; 1

2

x

xv 2

1

x

x

Bi tp 26: Cho ph-ng trnh2 (2 1) 0 x m x m (1)

a) CMR ph-ng trnh (1) lun c hai nghim phn bit vi mi m.b) Tm m ph-ng trnh c hai nghim tho mn : 1 2 1x x ;


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c) Tm m 2 21 2 1 26 x x x x t gi tr nh nht.Bi tp 27: Cho ph-ng trnh

2 2( 1) 2 10 0 x m x m (1)a) Gii ph-ng trnh vi m = 6.b) Tm m ph-ng trnh (1) c hai nghim 1 2,x x . Tm GTNN ca biu thc

2 21 2 1 210 A x x x x

Bi tp 28: Cho ph-ng trnh2( 1) (2 3) 2 0m x m x m (1)

a) Tm m (1) c hai nghim tri du.b) Tm m (1) c hai nghim 1 2,x x . Hy tnh nghim ny theo nghim kia.Bi tp 29: Cho ph-ng trnh

2 22( 2) ( 2 3) 0 x m x m m (1)

Tm m (1) c hai nghim 1 2,x x phn bit tho mn1 2

1 2

1 1

5

x x

x x

Bi tp 30: Cho ph-ng trnh2 0 x mx n c 3 2m = 16n.

CMR hai nghim ca ph-ng trnh , c mt nghim gp ba ln nghim kia.Bi tp 31 : Gi 1 2,x x l cc nghim ca ph-ng trnh

22 3 5 0x x . Khng gii ph-ng trnh ,

hy tnh : a)1 2

1 1x x

; b) 21 2( )x x ;

c) 3 31 2

x x d) 1 2x x

Bi tp 32 : Lp ph-ng trnh bc hai c cc nghim bng :

a) 3 v 2 3 ; b) 2 3 v 2 + 3 .Bi tp 33 : CMR tn ti mt ph-ng trnh c cc h s hu t nhn mt trong cc nghim l :

a)3 5

3 5

; b)

2 3

2 3

; c) 2 3

Bi tp 33 : Lp ph-ng trnh bc hai c cc nghim bng:

a) Bnh ph-ng ca cc nghim ca ph-ng trnh 2 2 1 0x x ;b) Nghch o ca cc nghim ca ph-ng trnh 2 2 0 x mx Bi tp 34 : Xc nh cc s m v n sao cho cc nghim ca ph-ng trnh

2 0 x mx n cng l m v n.Bi tp 35: Cho ph-ng trnh

2 32 ( 1) 0 x mx m (1)a) Gii ph-ng trnh (1) khi m = 1.b) Xc nh m ph-ng trnh (1) c hai nghim phn bit , trong mt nghim bng bnhphung nghim cn li.

Bi tp 36: Cho ph-ng trnh22 5 1 0x x (1)

Tnh 1 2 2 1 x x x x ( Vi 1 2,x x l hai nghim ca ph-ng trnh)

Bi tp 37: Cho ph-ng trnh2(2 1) 2 1 0m x mx (1)

a)Xc nh m ph-ng trnh c nghim thuc khong ( 1; 0 ).b) Xc nh m ph-ng trnh c hai nghim 1 2,x x tho mn

2 21 2 1x x


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Bi tp 38 :Cho phng trnh x2 - (2k - 1)x +2k -2 = 0 (k l tham s).

Chng minh rng phng trnh lun lun c nghim.

Bi tp 39:Tm cc gi r ca a ptrnh:

032)3( 222 axaxaa Nhn x=2 l nghim .Tm nghim cn li ca ptrnh?

Bi tp 40 Xc nh gi tr ca m trong ph-ng trnh bc hai :2 8 0 x x m

4 + 3 l nghim ca ph-ng trnh . Vi m va tm -c , ph-ng trnh cho cn mtnghim na . Tm nghim cn li y?

Bi tp 41: Cho ph-ng trnh : 2 2( 1) 4 0 x m x m (1) , (m l tham s).

1) Gii ph-ng trnh (1) vi m = 5.2) Chng minh rng ph-ng trnh (1) lun c hai nghim 1 2,x x phn bit mi m.

3) Tm m 1 2x x t gi tr nh nht ( 1 2,x x l hai nghim ca ph-ng trnh (1) ni trong phn 2/ ) .

Bi tp 42:

Cho phng trnh

1. Gii phng trnh khi b= -3 v c=22. Tm b,c phng trnh cho c hai nghim phn bit v tch ca chng bng 1

Bi tp 43:Cho phng trnh x2 2mx + m2 m + 1 = 0 vi m l tham s v x l n s.

a) Gii phng trnh vi m = 1.b) Tm m phng trnh c hai nghim phn bit x1 ,x2.

c) Vi iu kin ca cu b hy tm m biu thc A = x1 x2 - x1 - x2 t gi tr nh nht.Bi tp 44:

Cho ph-ng trnh ( n x) : x4 2mx2 + m2 3 = 01) Gii ph-ng trnh vi m = 3 2) Tm m ph-ng trnh c ng 3 nghim phn bit

Bi tp 45: Cho ph-ng trnh ( n x) : x2 2mx + m22

1= 0 (1)

1) Tm m ph-ng trnh (1) c nghim v cc nghim ca ptrnh c gi tr tuyt i bng nhau2) Tm m ph-ng trnh (1) c nghim v cc nghim y l s o ca 2 cnh gc vung ca mttam gic vung c cnh huyn bng 3.

Bi tp 46: Lp ph-ng trnh bc hai vi h s nguyn c hai nghim l:

53

41

x v

53

42

x

1) Tnh : P =44

53

4

53

4

Bi tp 47: Tm m ph-ng trnh: 0122 mxxx c ng hai nghim phn bit.
http://hocmai.vn/filter/tex/displaytex.php?x%5E2+%2B+bx+%2B+c+%3D+0http://hocmai.vn/filter/tex/displaytex.php?x%5E2+%2B+bx+%2B+c+%3D+0http://hocmai.vn/filter/tex/displaytex.php?x%5E2+%2B+bx+%2B+c+%3D+0


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12

Bi tp 48: Cho hai ph-ng trnh sau :2

2

(2 3) 6 0

2 5 0

x m x

x x m

( x l n , m l tham s )

Tm m hai ph-ng trnh cho c ng mt nghim chung.

Bi tp 49:

Cho ph-ng trnh : 2 22( 1) 1 0 x m x m vi x l n , m l tham s cho tr-c

1) Gii ph-ng trnh cho kho m = 0.2) Tm m ph-ng trnh cho c hai nghim d-ng 1 2,x x phn bit tho mn iu kin2 21 2 4 2x x

Bi tp 50: Cho ph-ng trnh :

22 1 2 3 0m x m x m ( x l n ; m l tham s ).

1) Gii ph-ng trnh khi m = 9

2

2) CMR ph-ng trnh cho c nghim vi mi m.3) Tm tt c cc gi tr ca m sao cho ph-ng trnh c hai nghim phn bit v nghim ny gpba ln nghim kia.

Bi tp 52: Cho ph-ng trnh x2 + x 1 = 0 .a) Chng minh rng ph-ng trnh c hai nghim tri du .

b) Gi 1x l nghim m ca ph-ng trnh . Hy tnh gi tr biu thc :81 1 110 13 P x x x

Bi tp 53: Cho ph-ng trnh vi n s thc x:x2 2(m 2 ) x + m 2 =0. (1)

Tm m ph-ng trnh (1) c nghim kp. Tnh nghim kp .

Bi tp 54:Cho ph-ng trnh : x2 + 2(m1) x +2m 5 =0. (1)a) CMR ph-ng trnh (1) lun c 2 nghim phn bit vi mi m.b) Tm m 2 nghim 1 2,x x ca (1) tho mn:

2 21 2 14x x .

Bi tp 55:a) Cho a = 11 6 2 , 11 6 2b . CMR a, ,b l hai nghim ca ph-ng trnh bc hai vi h snguyn.

b) Cho 3 36 3 10, 6 3 10c d . CMR 2,c d l hai nghim ca ph-ng trnh bc hai vi h snguyn.Bi tp 56: Cho ph-ng trnh bc hai :

2 22( 1) 1 0 x m x m m (x l n, m l tham s).1) Tm tt c cc gi tr ca tham s m ph-ng trnh c 2 nghim phn bit u m.2) Tm tt c cc gi tr ca tham s m ph-ng trnh c 2 nghim 1 2,x x tho mn : 1 2 3x x .

3) Tm tt c cc gi tr ca tham s m tp gi tr ca hm sy= 2 22( 1) 1 x m x m m cha on 2;3 .


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13

Bi tp 57:Cho ph-ng trnh :x2 2(m1) x +2m 3 =0.

a) Tm m ph-ng trnh c 2 nghim tri du.b) Tm m ph-ng trnh c 2 nghim ny bng bnh ph-ng nghim kia.

Bi tp 58: Cho ph-ng trnh : 2 26 6 0. x x a a 1) Vi gi tr no ca a th ph-ng trnh c nghim.2) Gi s

1 2,x x l nghim ca ph-ng trnh ny. Hy tm gi tr ca a sao cho 3

2 1 18 x x x

Bi tp 59: Cho ph-ng trnh :mx2 5x ( m + 5) = 0 (1) trong m l tham s, x l n.

a) Gii ph-ng trnh khi m = 5.b) Chng t rng ph-ng trnh (1) lun c nghim vi mi m.c) Trong tr-ng hp ph-ng trnh (1) c hai nghim phn bit 1 2,x x , hy tnh theo m gi tr ca

biu thc B = 2 21 2 1 210 3( ) x x x x . Tm m B = 0.

Bi tp 60:a) Cho ph-ng trnh : 2 22 1 0 x mx m ( m l tham s ,x l n s). Tm tt c cc gi tr nguynca m ph-ng trnh c hai nghim 1 2,x x tho mn iu kin 1 22000 2007x x

b) Cho a, b, c, d R . CMR t nht mt trong 4 ph-ng trnh sau c nghim2

2

2

2

2 0;

2 0;

2 0;

2 0;

ax bx c

bx cx d

cx dx a

dx ax b

Bi tp 61:1) Cho a, b , c, l cc s d-ng tho mn ng thc 2 2 2a b ab c . CMR ph-ng trnh

2

2 ( )( ) 0 x x a c b c c hai nghim phn bit.Cho ph-ng trnh 2 0 x x p c hai nghim d-ng 1 2,x x . Xc nh gi tr ca p khi

4 4 5 51 2 1 2 x x x x

t gi tr ln nht.Bi tp 62: Cho ph-ng trnh :(m + 1 ) x2 ( 2m + 3 ) x +2 = 0 , vi m l tham s.

a) Gii ph-ng trnh vi m = 1.b) Tm m ph-ng trnh c hai nghim phn bit sao cho nghim ny gp 4 ln nghim kia.

Bi tp 63: Cho ph-ng trnh: 2 23 2 2 10 4 0 x y xy x y (1)

1) Tm nghim ( x ; y ) ca ph-ng trnh ( 1 ) tho mn 2 2 10x y 2) Tm nghim nguyn ca ph-ng trnh (1).Bi tp 64: Gi s hai ph-ng trnh bc hai n x :

21 1 1 0a x b x c v

22 2 2 0a x b x c

C nghim chung. CMR

: 2

1 2 2 1 1 2 2 1 1 2 2 1 .a c a c a b a b b c b c

Bi tp 65: Cho ph-ng trnh bc hai n x :


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14

2 22( 1) 2 3 1 0 x m x m m a) Chng minh ph-ng trnh c nghim khi v ch khi 0 1m

b) Gi 1 2,x x l nghim ca ph-ng trnh , chng minh : 1 2 1 29

8x x x x

Bi tp 66: Cho ph-ng trnh bc hai n x :2 22 2 2 0 x mx m

a) Xc nh m ph-ng trnh c hai nghim.b) Gi

1 2,x x l nghim ca ph-ng trnh , tm gi tr ln nht ca biu thc :

1 2 1 22 4 A x x x x .

Bi tp 67: Cho ph-ng trnh bc hai n x :2( 1) 2( 1) 3 0m x m x m vi m 1. (1)

a) CMR (1) lun c hai nghim phn bit vi mi m.b) Gi 1 2,x x l nghim ca ph-ng trnh (1) , tm m 1 2 0x x v 1 22x x Bi tp 68: Cho a , b , c l di 3 cnh ca 1 tam gic . CMR ph-ng trnh

2 ( ) 0 x a b c x ab bc ac v nghim .Bi tp 69: Cho cc ph-ng trnh bc hai n x :

2

2

0(1);

0(2).

ax bx c

cx dx a

Bit rng (1) c cc nghim m v n, (2) c cc nghim p v q. CMR :2 2 2 2

4m n p q .Bi tp 70: Cho cc ph-ng trnh bc hai n x :

2 0 x bx c c cc nghim 1 2,x x ; ph-ng trnh2 2 0 x b x bc c cc nghim 3 4,x x .

Bit 3 1 4 2 1 x x x x . Xc nh b, c.Bi tp 71 : Gii cc ph-ng trnh saua) 3x4 5x2 +2 = 0b) x6 7x2 +6 = 0c) (x2 +x +2)2 12 (x2 +x +2) +35 = 0d) (x2 + 3x +2)(x2+7x +12)=24

e) 3x2+ 3x = xx 2 +1

f) (x + x1

) 4 ( )1

xx +6 =0

g) 121 2 xx

h) 20204 xx

i) (1048

3 2

2

x

x)

4

3 x

x

Bi tp 72. gii cc ph-ng trnh sau.

a) x2 5 x 5 =0 b) 5 .x2 2 x +1=0

c) ( 1 03)13()3 2 x d)5x4 7x2 +2 = 0 e) (x2 +2x

+1)2 12 (x2 +2x +1) +35 = 0 f) (x2 4x +3)(x212x +35)=16 g) 2x2+ 2x = xx 2

+1 .Bi tp 73.Cho ph-ng trnh bc hai 4x25x+1=0 (*) c hai nghim l x 1 , x 2 .

1/ khng gii ph-ng trnh tnh gi tr ca cc biu thc sau:

22

21

11

xxA ; B

22

22

1

1 44

x

x

x

x

; 52

51 xxC ;

72

71 xxD

2/ lp ph-ng trnh bc hai c cc nghim bng:a) u = 2x1 3, v = 2x23


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15

b) u =1x

1

1 , v =

1x

1

2 .

Bi tp 74 . Cho hai ph-ng trnh : x2 mx +3 = 0 v x2 x +m+2= 0 .a) Tm m ph-ng trnh c nghim chung.b) Tm m hai ph-ng trnh t-ng -ng.Bi tp 75. Cho ph-ng trnh (a3)x2 2(a1)x +a5 = 0 .a) tm a ph-ng trnh c hai nghim phn bit x1, x2.

b) Tm a sao cho 1x

1

+ 2x

1


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16

a) Gii PT khi m = 2b) C/mr phg-ng trnh cho c hai nghim tri du vi mi GT ca mc) Gi hai nghim cu PT cho l x1 ; x2 .Tm m hai nghim tho mn

3 3

1 2

2 1

x x

x x

t GTLN

Bi tp 88: Cho Ph-ng trnh : x2 mx m 1 = 0 (*)a) C/mr PT (*) c nghim x1 ; x2 vi mi GT ca m ; tnh nghim kp ( nu c ) ca PT v GT m t-ng -ng.b) t A = x12 + x22 6x1.x21) Chng minh A = m2 8m + 82) Tm m sao cho A= 83) Tm GTNN ca a v GT m t-ng ng .Bi tp 89: Cho ph-ng trnh x2 2(a 1) x + 2a 5 = 0 (1)a) C/mr PT(1) c nghim vi mi ab) Vi gi tr no ca a th (1) c nghim x1 ,x2 tho mn x1 < 1 < x2c) Vi gi tr no ca a th ph-ng trnh (1) c hai nghim x1, x2 tho mn

x12 + x2

2 =6Bi tp 90: Cho ph-ng trnh : x2 2(m+1)x + m 4 = 0 ( *)a) Chng minh (*) c hai nghim vi mi mb) Tm gi tr ca m PT (*) c hai nghim tri duc) Gi s x1 ; x2 l nghim ca PT (*)

Chn minh rng : M = (1 x1) x2 + (1 x2)x1Bi tp 91: Cho ph-ng trnh : x2 (1 2n) x + n 5 = 0a) Gii PT khi m = 0b) Chng minh rng PT c nghim vi mi gi tr ca nc) Gi x1; x2 l hai nghim cu PT choChng minh rng biu thc : x1(1 + x2) + x2(1 +x1)Bi tp 92: Cc nghim ca ph-ng trnhx2 + ax + b + 1 = 0 (b khc 1) l nhng s nguynChng minh rng a2 + b2 l hp s

Bi tp 93: Cho a,b,c l ba cnh ca tam gic .C/m:x2 + ( a + b + c) x + ab + bc + ca = 0

v nghim

Bi tp 94: Cho cc ph-ng trnh ax2 + bx + c = 0 ( a.c 0) v cx2 + dx + a = 0 c cc nghim x1; x2 v y1 ;y2 t-ng -ng C/m x1

2 + x22 + y1

2 + y22 4

Bi tp 95: Cho cc ph-ng trnh x2+ bx +c =0 (1) v x2 +cx +b = 0 (2)

Trong 2

111

cb

Bi tp 96: Cho p,q l hai s d-ng .Gi x1 ; x2 l hai nghim ca ph-ng trnhpx2 + x +q = 0 v x3 ; x4 l nghim ca ph-ng trnh qx

2 + x + p = 0

C/m : 1 2 3 4. . 2 x x x x

Bi tp 97: Cho a,b,c l ba s thc bt k .Chng minh rng t nht mt trong ba ph-ng trnh sau c nghim

:2 2 21 0; 1 0; 1 0 x ax b x bx c x cx a

Bi tp 98: Cho ph-ng trnh bc hai :x2 + (m+2) x + 2m = 0 (1)a) C/m ph-ng trnh lun lun c nnghimb) Gi x1; x2 l hai nghim ca ph-ng trnh . Tm m 2(x1

2 + x22 ) = 5x1x2

Bi tp 99: Cho ph-ng trnh x2 + a1x + b1 = 0 (1) ;x2 + a2x + b2 = 0 (2)

C cc h s tho mn 1 2 1 22a a b b .Cmr t nht mt trong hai ph-ng trnh trn c nghim


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17

Bi tp 100: Chng minh rng ph-ng trnh : 2 2 2 2 2 2 0a x b a c x b

V nghimNu a + b > c v a b c

Bi tp 101: Cho hai ph-ng trnh :x2 + mx + 1 = 0 (1) x2 + x + m = 0 (2)

a) Tm m hai ph-ng trnh trn c t nht mt nghim chung

b) Tm m hai ph-ng trnh trn t-ng -ng

Bi tp 102: Cho ph-ng trnh:x2 2( a + b +c) x + 3( ab + bc+ ca) = 0 (1)a) C/mr ph-ng trnh (1) lun c nghimTrong tr-ng hp ph-ng trnh (1) c nghim kp xc nh a,b,c .Bit a2 + b2 + c2 = 14Bi tp 103: Chng minh rng nu ph-ng trnh :x2 + ax + b = 0 v x2 + cx + d = 0 c nghim chung th : (b

d)2 + (a c)(ad bc) = 0Bi tp 104: Cho ph-ng trnh ax2 + bx + c = 0 .C/mr nu b > a + c th ph-ng trnh lun c 2 nghim phnbitBi tp 105: G/s x1 , x2 l hai nghim ca hai ph-ng trnh x

2 + ax + bc = 0 v x2 , x3 l hai nghim caph-ng trnh x2 + bx + ac = 0 ( vi bc khc ac ) . Chng minh x1, x3 l nghim ca ph-ng trnh x

2 + cx + ab =

0 .Bi tp 106: Cho ph-ng trnh x2 + px + q = 0 (1) .Tm p,q v cc nghim ca ph-ng trnh (1) bit rng khithm 1 vo cc nghim ca n chng ch thnh nghim ca ph-ng trnh : x2 p2x + pq = 0Bi tp 107: Chng minh rng ph-ng trnh :

(x a) (x b) + (xc) (x b) + (xc) (x a) = 0Lun c nghim vi mi a,b,c.

Bi tp 108: Gi x1; x2 l hai nghim ca ph-ng trnh : 2x2 + 2(m +1) x + m2 +4m + 3 = 0

Tm GTLN ca biu thc A = 1 2 1 22 2 x x x x

Bi tp 109: Cho a 0 .G/s x1 ; x2 l nghim ca ph-ng trnh2

2

10

2 x ax

a

Chng minh rng : 4 41 2 2 2x x

Bi tp 110 Cho ph-ng trnh 22

1 0 x axa

.Gi x1 ; x2 l hai nghim ca ph-ng trnh

Tm GTNN ca E = 4 41 2x x

Bi tp 111: Cho pt x2 + 2(a + 3) x + 4( a + 3) = 0a) Vi gi tr no ca a th ph-ng trnh c nghim kpb) Xc nh a ph-ng trnh c hai nghim ln hn 1


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18

H ph-ng trnh bc nht hai n

Dng

''' cybxa

cbyax

1. Gii h phng trnh

1)

3)12(412)12(

yxyx 2)

53

1

7

3

1

3

2

5

3

yx

yx

2. Gii v bin lun h phng trnh

1)

55

55

myx

ymx2)

mmyxm

myxm

3)1(

72)5(

3. Tm gi tr ca tham s h phng trnh c v s nghim

1)

23)12(

3)12(

mmyxm

mymmx2)

mnmynx

nmnymx

2

22

4. Tm m hai ng thng sau song songmy

mxmyx

1)1(,046

5. Tm m hai ng thng sau ct nhau trn Oymymxmmyx 3)32(,2

H gm mt ph-ng trnh bc nht vmt ph-ng trnh bc hai hai n

Dng

)2(

)1(22 khygxeydxycx

cbyax

PP gii: Rt xhoc y (1) ri th vo (2).1. Gii h phng trnh

1)

423

53222 yyx

yx2)

5)(3

0143

yxxy

yx

3)

100121052

13222 yxyxyx

yx

2. Gii v bin lun h phng trnh

1)

22

1222 yx

ymx2)

22

1222 yx

ymx

3. Tm m ng thng 0)1(88 mymx

ct parabol 02 2 xyx ti hai im phn bit.

H ph-ng trnh i xng loi I

Dng

0),(

0),(

2

1

yxf

yxf; vi ),( yxfi = ),( xyfi .


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19

PP gii: t PSPxy

Syx4; 2

1. Gii h phng trnh

1)

7

522 xyyx

xyyx2)

30

1122 xyyx

xyyx

3)

931

19

2244

22

yxyx

xyyx 4)

243

2111

33 yx

yx

5)

491

1)(

51

1)(

2222

yxyx

xyyx

6)

2

5

1722

y

x

y

x

yx

2. Tm m h phng trnh c nghim

1)

myx

yx

66

22 12)

mxyyx

yxyx

)1)(1(

8)22

3. Cho h phng trnh

32

22 xyyx

myx

Gi s yx; l mt nghim ca h. Tm m biu thc F= xyyx 22 t max, t min

H ph-ng trnh i xng loi II

Dng

0),(

0),(

xyf

yxf

PP gii: h tng ng

0),(),(

0),(

xyfyxf

yxf

hay

0),(),(0),(),(

xyfyxf

xyfyxf

1. Gii h phng trnh

1)

yxx

xyy

43

432

2

2)

yxyx

xxyy

3

32

2

3)

yxyx

xyxy

40

4023

23

4)

yxx

xyy

83

833

3

2. Tm m h phng trnh c nghim duy nht.

1)

myxx

myxy

2)(

2)(

2

2

2)

myyyx

mxxxy

232

232

4

4

H ph-ng trnh ng cp (cp 2)

Dng

)2(''''

)1(22

22

dycxybxa

dcybxyax

PP gii: t tx nu 0x 1. Gii h phng trnh


20/34

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20

1)

932

22222

22

yxyx

yxyx2)

42

133222

22

yxyx

yxyx

3)

16

1724322

22

yx

yxyx4)

137

152

22

xyy

yx

2. Tm m h phng trnh c nghim

1)

myxyx

yxyx

1732

1123

22

22

2)

myxyx

yxyx

22

22

54

132

Mt s H ph-ng trnh khc1. Gii h phng trnh

1)

7

122 yxyx

yx2)

180

4922 xyyx

xyyx

3)

7

2)(33 yx

yxxy4)

0)(9)(8

01233 yxyx

xy

5)

21

122

yx

yx6)

yxyx

xyxy

10)(

3)(2

22

22

2. Gii h phng trnh

1)

12

527

yxyx

yxyx3)

7

142

222

zyx

yxz

zyx

2)

523

53

2323 22

yx

xxyy

3. Tm m hai phng trnh sau c nghim chung

a) mx 31 v 12422

mx b) 01)2()1( 2 xmxm v

0122 mxx 4. Tm m h phng trnh c nghim

02

)1(

xyyx

xyayx

11

1

xy

myx

4. Tm m, n h phng trnh sau c nhiuhn 5 nghim phn bit

myxyyxmx

ynxyx

22

22

)(

1

PHN 5BT NG THC

Dng nh ngha

Chng minh cc bt ng thc sau


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21

1.Cho a,b,c,d > 0

a) nu a < b tha

b b tha

b >a + c

b + c

c) 1 b > 0 v c ab . Chng minh rngc + ac2 + a2

c + bc2 + b2

5.Cho a + b + c 0. Chng minh rng :a3 + b3 + c3 3abc

a + b + c 0

5.Cho ba s dng a ,b ,c ,chng minh rng :1

a3 + b3 + abc +1

b3 + c3 + abc +1

c3 + a3 + abc 1

abc

4.Cho cc s a,b,c,d tho a b c d 0. Chng minh rng :

a) a2 b2 + c2 (a b + c)2 b) a2 b2 + c2 d2 (a b + c d)2

5.a) Cho a.b 1,Chng minh rng :1

1 + a2 +1

1 + b2 2

1 + ab

a) Cho a 1, b 1 .Chng minh rng :1

1 + a3 +1

1 + b3 +1

1 + c3 3

1 + abc

b) Cho hai s x ,y tho x + y 0.Chng minh rng :1

1 + 4x +1

1 + 4y 2

1 + 2x+y

6. a,b,c,d chng minh rnga) a2 + b2 + c2 + d2 (a + c)2 + (b + d)2


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22

b) 1 a3 + b3 + c3d) a3(b2 c2) + b3(c2 a2) + c3(a2 b2) < 0e) (a + b + c)2 9bc vi a b cf) (a + b c)(b + c a)(c + a b) abc8. Cho hai s a ,b tho a + b 2 ,chng minh rng : a4 + b4 a3 + b39.Cho a ,b ,c 0 , chng minh rng :a) a3 + b3 + c3 3abc

b) a3b + b3c + c3a a2bc + b2ca + c2abc) a3(b2 c2) + b3(c2 a2) + c3(a2 b2) < 010. Cho a ,b ,c l di 3 cnh mt tam gic,vi a b c

Chng minh rng : (a + b + c)2 9bc

*.Cho tam gic ABC,chng minh rng :aA + bB + cC

a + b + c 3

*.Cho a ,b ,c [0;2] . Chng minh rng : 2(a + b + c) (ab + bc + ca) 4. Chng minh rng :

11.2 +

12.3 +

13.4 + +

1n(n + 1) < 1 n N

. Chng minh rng :12! +

23! +

34! + +

n 1n! < 1 n N n 2

*.Cho ba s dng a ,b ,c tho mn: ab + bc + ca = 1 . Chng minh rng :

3 a + b + c 1

abc

.11.Cho 3 s a, b, c tho mn a + b + c = 3. Chng minh rng :a) a2 + b2 + c2 3

b) a4 + b4 + c4 a3 + b3 + c3

Bt ng thc Cauchy1.Cho hai s a 0 , b 0 Chng minh rng :

a)a

b +ba 2 a , b > 0 b) a

2b +1

b 2a b > 0 c)2a2 + 14a2 + 1

1

d) a3 + b3 ab(a + b) e) a4 + a3b + ab + b2 4a2bf) (a + b)(1 + ab) 4ab g) (1 + a)(1 + b) (1 + ab )2

h)a2

a4 + 1 12 i)

1a +

1b

4a + b j)

1a +

1b +

1c

2a + b +

2b + c +

2c + a

j) (1 + a)(1 + b) (1 + ab )2 h)a2 + 2a2 + 1

2 k)a6 + b9

4 3a2b3 16

l) a

2

+ 6a2 + 2 4 m) a

2

b2 +b

2

c2 + c

2

a2 ac + cb +ba

2.Cho a > 0 , chng minh rng : (1 + a)2

1a2 +

2a + 1 16

3. Cho 3 s a ,b ,c > 0 ty . Chng minh rng:

a) a2b +1

b 2a b) a + b + c 12 ( a

2b + b2c + c2a +1a +

1b +

1c )


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4.Cho 0 < a < b , chng minh rng: a b v ab = 1 ,chng minh rng :a2 + b2

a b 2 2

Chng minh rng 12

(a + b)(1 ab)(1 + a2)(1 + b2)

12

13 .a) Chng minh rng nu b > 0 , c > 0 th :b + c

bc 4

b + c


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b)S dng kt qu trn chng minh rng nu a ,b ,c l ba s khng m c tnga + b + c = 1 th b + c 16abc

14.Cho a + b = 1,Chng minh rng: (1 +1a )(1+

1b ) 9

15.Cho a,b,c > 0 v a + b + c = 1 . Chng minh rng :

a) (1 +1a )(1+

1b )(1+

1c ) 64 b) (a + b)(b + c)(c + a)abc

8729

16.Cho 4 s a ,b ,c ,d > 0 tho mn1

1 + a+

1

1 + b+

1

1 + c+

1

1 + d 3

Chng minh rng abcd 181

17.Cho a,b,c l di ba cnh ca mt tam gic ,chng minh rng :a) ab + bc + ca < a2 + b2 + c2 < 2(ab + bc + ca)

b) abc (a + b c)(b + c a)(c + a b)

c) (p a)(p b)(p c) abc8 d)

1p a +

1p b +

1p c 2(

1a +

1b +

1c )

e) p < p a + p b + p c < 3p18.Cho 3 s a ,b ,c 0 ,tho mn a.b.c = 1

Chng minh rng : (1 + a)(1 + b)(1 + c) 819. Cho 3 s x, y, z tho mn: x2 + y2 + z2 = 1. Chng minh rng

1 x + y + z + xy + yz + zx 1 + 320 .Cho n s dng a1 ,a2 ,.,an. Chng minh rng

a)a1a2

+a2a3

+ +ana1

n b) (a1 + a2 + + an)(1a1

+1a2

+ +1an

) n2

c) (1 + a1)(1 + a2)(1 + an) 2n vi a1.a2.an = 1

21.Cho n s a1 ,a2 ,.,an [0;1] ,chng minh rng :(1 + a1 + a2 + + an)

2 4(a12 + a2

2 + + an2)

22.Cho a > b > 0 , chng minh rng : a +1

b(a b) 3 .Khi no xy ra du =

23. Cho hai s a 0 ; b 0 . Chng minh rng :

a) 2 a + 33

b 55

ab b)17125

ab17b12a5 c)

a6 + b9

4 3a2b3 16

24. Chng minh rng 1.3.5.(2n 1) < nn25.Cho ba s khng m a ,b ,c chng minh rng :

a + b + c knm nmkknm mknknm knm cbacbacba 26 .Cho 2n s dng a1 ,a2 ,.,an v b1 ,b2 ,.,bn.

Chng minh rng :n

a1.a2....an +n

b1.b2....bn n

(a1 + b1)(a2 + b2).(an + bn)

27. Chng minh rng :4

(a + 1)(b + 4)(c 2)(d 3)

a + b + c + d

1

4

a 1 , b 4 , c 2 ,d > 328*. n N chng minh rng :

a) 1.122

.133

. 144

.. 1nn ( 1 +1n )

n


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25

30*.Cho x1,x2,xn > 0 v x1 + x2 + .+ xn = 1 Chng minh rng

(1 +1x1

)(1+1x2

)(1+1xn

) (n + 1)n

31*.Cho cc s x1,x2 ,y1,y2,z1,z2 tho mn x1.x2 > 0 ; x1.z1 y12 ; x2.z2 y2

2Chng minh rng : (x1 + x2)(z1 + z2) (y1 + y2)

2

32*.Cho 3 s a ,b ,c (0;1). Chng minh rng trong 3 bt ng thc sau phi c mt bt ng thc sai:a(1 b) > 1/4 (1) ; b(1 c) > 1/4 (2) ; c(1 a) > 1/4 (3)

33*.Cho 3 s a,b,c > 0. Chng minh rng :

2 aa3 + b2 + 2 bb3 + c2 + 2 cc3 + a2 1a2 + 1b2 + 1c2

34** Cho x ,y ,z [0;1] ,chng minh rng : (2x + 2y + 2z)(2 x + 2 y + 2 z) 818

(HBK 78 trang 181,BT Trn c Huyn)35*.Cho a , b , c > 1. Chng minh rng :

a) log2a + log2b 2 log2

a + b2

b) 2

logbaa + b +

logcbb + c +

logacc + a

9a + b + c

36*Cho a ,b ,c > 0,chng minh rng :

a) ab + c + bc + a + ca + b 32 b) a

2

b + c + b

2

c + a + c

2

a + b a + b + c2

c)a + b

c +b + c

a +c + ab 6 d)

a3

b +b3

c +c3

a ab + bc + ca

e) (a + b + c)(a2 + b2 + c2) 9abc f)bc

a +acb +

abc a + b + c

g)a2

b + c +b2

c + a +c2

a + b a + b + c

2 ab

a + b +bc

b + c +ca

c + a

37.Cho ba s a ,b ,c tu . Chng minh rng :a2(1 + b2) + b2(1 + c2) + c2(1 +ab2) 6abc

38*Cho a ,b ,c > 0 tho :1a +

1c =

2b . Chng minh rng :

a + b2a b +

c + b2c b 4

39*Cho 3 s a, b, c tho a + b + c 1. Chng minh rng :

a)1a +

1b +

1c 9 b)

1a2 + 2bc +

1b2 + 2ac +

1c2 + 2ab 9

40*Cho a ,b ,c > 0 tho a + b + c k. Chng minh rng :

(1 +1a )(1 +

1b )(1 +

1c ) (1 +

3k)

3

41*Cho ba s a ,b ,c 0. Chng minh rng :a2

b2 +b2

c2 +c2

a2 a

b +bc +

ca

42*Cho tam gic ABC,Chng minh rng :

a) ha + hb + hc 9r b)a ba + b +

b cb + c +

c ac + a 0a) x2 + 4y2 + 3z2 + 14 > 2x + 12y + 6z

b) 5x2 + 3y2 + 4xy 2x + 8y + 9 0c) 3y2 + x2 + 2xy + 2x + 6y + 3 0d) x2y4 + 2(x2 + 2)y2 + 4xy + x2 4xy3e) (x + y)2 xy + 1 3 (x + y)


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26

f) 3

x2

y2 +y2

x2 8

xy +

yx + 10 0

g) (xy + yz + zx)2 3xyz(x + y + z)2.Cho 4 s a ,b ,c ,d tho b< c < d chng minh rng :

(a + b + c + d)2 > 8(ac + bd)3. Chng minh rng : (1 + 2x + 3x)2 < 3 + 3.4x + 32x+14. Cho ax + by xy , x,y > 0. Chng minh rng : ab 1/4

*5. Cho 1 x 1

2v

5

6< y 0

6** Cho a3 > 36 v abc = 1.Xt tam thc f(x) = x2 ax 3bc +a2

3

a) Chng minh rng : f(x) > 0 x

b) Chng minh rng:a2

3 + b2 + c2 > ab + bc + ca

Cho hai s x , y tho mn: x y . Chng minh rng x3 3x y3 3y + 4.Tm Gi tr nh nht ca cc hm s :

a) y = x2 +4x2

b) y = x + 2 +1

x + 2

vi x > 2

c) y = x +1

x 1 vi x > 1

d) y =x3 +

1x + 2 vi x > 2

e) y =x2 + x + 1

x vi x > 0

f) y =4x +

91 x vi x (0;1)

.Tm gi tr ln nht ca cc hm s sau:y = x(2 x) 0 x 2

y = (2x 3)(5 2x)

3

2 x

5

2

y = (3x 2)(1 x)23 x 1

y = (2x 1)(4 3x)12 x

43

y = 4x3 x4 vi x [0;4].Trong mt phng ta Oxy,trn cc tia Ox v Oy ln lt ly cc im A v B thay i sao cho ngthng AB lun lun tip xc vi ng trn tm O bn knh R = 1. Xc nh ta ca A v B onAB c di nh nht*.Cho a 3 ; b 4 ; c 2 .Tm gi tr ln nht ca biu thc

A =ab c 2 + bc a 3 + ca b 4

abc

*Tm gi tr ln nht v gi tr nh nht ca hm s y = x 1 + 5 x

PHN 5

1. BT PHNG TRNH V H BT PHNG TRNH BC NHT MT N


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27

Bi 1. Gii cc bt phng trnh sau

2

3 1 2 1 2.

2 3 4

. (2 1)( 3) 3 1 ( 1)( 3) 5

x x xa

b x x x x x x

2

2 2

. 8 3

3. 1 2( 3) 5 4

2

c x x

d x x x

Bi 2 Gii cc h bpt sau:

a.3 1 2 7

4 3 2 19

x x

x x

b.

56 4 7

78 3

2 52

x x

xx

2. DU CA NH THC BC NHTBi 1. Xt du cc biu thc sau:

. ( ) (2 1)( 3)

4 3. ( )

3 1 2

a f x x x

b f xx x

2

. ( ) (3 2)( 2)( 3)

. ( ) 9 1

c f x x x x

d f x x

Bi 2 Gii cc bpt sau:

2

3 7.

2 2 11 1

.2 ( 2)

ax x

bx x

2

2

1 2 3.

3 2

3 3. 1

4

c x x x

x xd

x

2

3. 1

2

2. 1

1 2

ex

x xf x

x

Bi 3. Gii cc bpt sau:. 2 1 3 5

. 3 1 2

. 3 2 2

a x x

b x x

c x x

.2 1 1d x x

3.DU CA TAM THC BC HAIBi 1. Xt du cc biu thc sau:

2 2

2 2

2 2

. ( ) 3 2 . ( ) 2 5 2

. ( ) 9 24 16 . ( ) 3 5

. ( ) 2 4 15 . 4 4

a f x x x b f x x x

c f x x x d f x x x

e f x x x f f x x x

Bi 2. Lp bng xt du cc biu thc sau:2

2 2

2 2

2 2

2

. ( ) (3 10 3)(4 5)

. ( ) (3 4 )(2 1)

. ( ) (4 1)( 8 3)(2 9)

(3 )(3 ). ( )4 3

a f x x x x

b f x x x x x

c f x x x x x

x x xd f x

x x

7.BT PHNG TRNH BC HAIBi 1. Gii cc bpt sau:


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28

22 2

2 2

1 3.4 1 0 .

4 3 4

. 3 4 0 . 6 0

a x x c x x x

b x x d x x

8.MT S PHNG TRNH V BT PHNG TRNH QUY V BC HAIGii cc bt phng trnh sau:

1)2 2 5

34

x xx

x

2)2 3 1

2

x xx

x

3)3 47 4 47

3 1 2 1

x x

x x

4)9

42

x

x

5)

3 4

3 2

1 2 60

7 2

x x x

x x

6)

24 2 4 2 x x x 7) 2 7 10 0x x

8) 2 23 2 5 6 0 x x x x 9)2 3

01 2

x x

x

10)

1

2+

2

2

3 2 1

4 3 3

x x x

x x x

11)2

2

2 3 4 15

1 1 1

x x x x

x x x

12)

2

2 1 42 2 2 x x x

13)

2 3

1 2 2 31 1 1

x

x x x x

14)4 3 2

2

3 20

30

x x x

x x

15)

3 23 30

2 x x x

x x

16)

4 2

2

4 30

8 15

x x

x x

17)

3 4 5

21 2 3 6 0

7 x x x x

x x

18) 2

4211

x xx x

19) 22

2151

1x x

x x

Gii h bt phng trnh sau:

2)

2 31

12 2 4

01

x

x

x x

x

3)2 12 0

2 1 0

x x

x

4)

2

2

3 10 3 0

6 16 0

x x

x x

5)2

2

4 7 0

2 1 0

x x

x x

6)

2

2

5 0

6 1 0

x x

x x

7)

2

2

3 8 3 0

17 7 6 0

x x

x x

8)

2

2

2

4 3 02 10 0

2 5 3 0

x x

x x

x x

9)2

2

2 74 1

1

x x

x

10)

2

2

1 2 21

13 5 7

x x

x x

11)2

2

10 3 21 1

3 2

x x

x x

12)

2

2

2

3 40

3

2 0

x x

x

x x

13)

2

2

4 2

2

30

12 0

2 0

4 5 0

x x

x

x

x x

x x

Phng trnh v bt phng trnh c cha tr tuyt i:

1) 2 5 4 4 x x x 2) 2 22 8 1 x x x 3) 2 5 1 1 0x x

4) 31 1 x x x 5) 2 1 2 0x x 6) 1 4 2 1x x

7) 2 23 2 2 x x x x 8) 2 5 7 4x x 9)2

2

41

2x x

x x

10)2

2

5 41

4

x x

x

11)

2 51 0

3x

x

12)

2

23

5 6

x

x x


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29

13)2

2x x

x

14) 2

2

21x

x 15)

2

2

4 31

5

x x

x x

16) 2 3 3x x 17)

2 1 12

2

x x

x x

18) 2 4 2 x x x

19) 3 1 2x x 20)2

2

2 41

2

x x

x x

21) 1 3 x x x x

22)2 6

22

x xx

x

23) 2 1 5x x 24) 1 2 x x x

Phng trnh v bt phng trnh c cha cn :

1) 2 2 4 2 x x x 2) 23 9 1 2 x x x 3) 2 12 7 x x x

4) 221 4 3 x x x 5) 21 2 3 5 0 x x x 6) 2 1

2 12

xx

x

7)

2 16 533 3

xxx x

8)2

8 12 4 x x x 9)2 4 3

2x x

x

10) 2 22 2 4 3 x x x x 11) 21 2 3 4 x x x x 12) 2 23 12 3 x x x x

13) 26 2 32 34 48 x x x x 14) 23 6 3 x x x x

15) 24 1 3 5 2 6 x x x x 16) 2 24 6 2 8 12 x x x x

17) 22 1 1 1 x x x x 18) 2 23 5 7 3 5 2 1 x x x x 19) 2 22 4 4 x x x

20) 2

2

3 4 92 3

3 3

xx

x

21) 2 23 4 9 x x x 22)

2

2

9 43 2

5 1

xx

x

23)6 3 3

4 4 2 x x x 24) 3 4 1 8 6 1 1 x x x x

25) 26 9 6 9 1 x x x x 26) 1 2 3 x x x 27)4 1 3

1 4 2x x

x x

28)1 1 1

1x

x x x x

* tm tp xc nh ca mi hm s sau:

1) 2 3 4 8 y x x x 2)2 1

2 1 2x x

yx x

3)

2 2

1 1

7 5 2 5y

x x x x

4) 2 5 14 3 y x x x 5)2

3 31

2 15

xy

x x

Cc dng ton c cha tham s:Bi1: Tm cc gi tr ca m mi biu thc sau lun dng:

a) 2 4 5 x x m b) 2 2 8 1 x m x m c) 22 4 2 x x m

d) 23 1 3 1 4m x m x m e) 21 2 1 3 2m x m x m

Bi 2: Tm cc gi tr ca m mi biu thc sau lun m:a) 24 1 2 1m x m x m b) 22 5 4m x x c) 2 12 5mx x


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30

d) 24 1 1 x m x m e) 2 22 2 2 1 x m x m f) 22 2 3 1m x m x m

Bi 3: Tm cc gi tr ca tham s m mi bt phng trnh sau nghim ng vi mi gi tr x:

a) 21 2 1 3 3 0m x m x m b) 2 24 5 2 1 2 0m m x m x

c)

2

2

8 200

2 1 9 4x x

mx m x m

d)

2

2

3 5 40

4 1 2 1x x

m x m x m

Bi 4: Tm cc gi tr ca m phng trnh:

a)

2

2 1 9 5 0 x m x m

c hai nghim m phn bitb) 22 2 3 0m x mx m c hai nghim dng phn bit.

c) 25 3 1 0m x mx m c hai nghim tri duBi 5: Tm cc gi tr ca m sao cho phng trnh :

4 2 21 2 1 0 x m x m

a) v nghim b) C hai nghim phn bit c) C bn nghim phn bitBi 6 : Tm cc gi tr ca m sao cho phng trnh: 4 2 21 1 0m x mx m c ba nghim phn bit

Bi 7: Cho phng trnh: 4 22 2 1 2 1 0m x m x m . Tm cc gi tr ca tham s m pt trn c:

a) Mt nghim b) Hai nghim phn bit c) C bn nghim

Bi 8: Xc nh cc gi tr ca tham s m mi bt phng trnh sau nghim ng vi mi x:a)

2

2

11

2 2 3

x mx

x x

b)

2

2

2 44 6

1

x mx

x x

c)

2

2

51 7

2 3 2

x x m

x x

Bi 9: Tm cc gi tr ca tham s m bt phng trnh sau v nghim:2 10 16 0

3 1

x x

mx m

Bi 10: Tm cc gi tr ca tham s m bt phng trnh sau c nghim:

a)

2 2 15 0

1 3

x x

m x

b)

2 3 4 0

1 2 0

x x

m x

PHNG TRNH, BPT V T

Bi 1. Gii cc pt sau:1) x 1 8 3x 1 22) x 2 4 2-xx

23) 3x 9 1 x-2x 24) 3x 9 1 x-2x

5) 3x 7- x 1 2 2 26) x 5 x 8 4 5x x Bi 2. Gii cc bpt sau:

21) x 12 7-xx 22) 21-4x-x x 3 23) 1-x 2x 3 5 0x 24) x 3 10 x-2x

25) 3 -x 6 2(2x-1) 0x 26) 3x 13 4 2-x 0x

7) x 3- 7-x 2x-8 8) 2x 3 x 2 1 29) 2x x 1 x 1 10) 2-x 7-x - -3-2x

11) 11-x - x-1 2 4

12) - 2-x 22-x

2x 16 513) x-3

3 x-3x

14) 1-4x 2x 1


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32

69) 4 2 4 2 2x +x 1 + x x +1 2x 70) x+3 x1< x2 71) x+1 x1 x 72) 5x+1 4x1 3 x 73) x+1 > 3 x+4 74) x+2 3x< 52x

75) 2 2 2x +x+1+ x x+1 2x +6x+2 76) 6x + 1 2x + 3 < 8x 4x + 2

77) x + x + 9 x + 1 + x + 4 78) 3 312 x + 14 + x 2

79)3 3

4 x + x + 8 2 80)2

x 1 x < 0 81)

2

2

9x 40

5x 1 82) 22x 5 2x 5x + 2 0

83) 2 2(x 4x + 3) x 4 >0 84)2

(x1)x(x + 2) 0

(x2)

85) 2 2(x 3x) 2x 3x 2 0 86) 2 2( 2) x + 4 x 4x

87)2

2

3(4x 9)2x+3

3x 3 88) 2 2(x 3) x + 4 x 9

89)2

29x 4 3x+25x 1

90) 2 2x(x 4) 4x x 4 (2 x)

91)2x

3x 2 1 x3x 2

92)2

2 2

2

xx x 4+ 4x

2 4 x

93) x+34x+1 3x25

94) 2

2 2

2

x3x 2x +1 25 x

5 + 25 x

95) 22

40x + x +16

x +16 96) 2 23x +5x+7 3x +5x+2 >1

97)2

24x < 2x + 9(1 1 + 2x )98) 2x > 2x + 22x + 1 1

99) 2 24(x + 1) < (2x + 10)(1 3 + 2x) 100)2

2

2xx + 21

(3 9 + 2x )

101)2

2

x> x 4

(1 + 1 + x )102) 2 29(x + 1) (3x + 7)(1 3x + 4)

103) (x1) 2x 1 3(x1) 104)2x

> 2x + 22x + 1 1

105)

2

2

x

> x 4(1 + 1 + x ) 106)

2

2

4x

< 2x + 9(1 1 + 2x )

107)2

2

2xx + 21

(3 9 + 2x ) 108) 2 24(x + 1) < (2x + 10)(1 3 + 2x)

109) 2 2x + 4x (x + 4) x 2x + 4 110) 2 29(x + 1) (3x + 7)(1 3x + 4)


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33

PHNG PHP I BIN S:Bi 1. Gii cc pt sau:

2 21) 3x 5 8- 3x 5 1 1x x 2 22) x 9- x 7 2

21 x 21 213)

x21 x 21

x

x

2 2 24) 3x 6 16 2 2 x 2 4 x x x x

5) (x 5)(x-2) 3 x x 3 0 6) 2(x 1)(x 4) 5 x 5 28x

7) 2 23x 5 7- 3x 5 2 1x x 8) 2(x 4)(x 1)-3 x 5 2 6x 9) 24 4 x 2 x x 2x 8 10) 2 22x x 5x 6 10x 15

11) 2 22x 4x 3 3 2x x 1 12) 26 (x 2)(x 32) x 34x+48

13) 2x(x + 3) 6 x 3x 14) 2(x +4)(x +1) 3 x +5x+2 x x+1

17) 2(x +1)(x +4) < 5 x +5x+28 18) 2 2x + 2x + 5 4 2x + 4x+3

19) 2 22x + x 5x 6 >10x+15 20) 4x x1 3 >

x1 4x 2

21) x x+1 2 >3x+1 x

22) 46x 12x 12x 2. 0x2 x2 x2

23) 3 6x2 x2 x2+2. + 4 0x+1 x+1 x+1

24)5 1

5 x+ < 2x+ +42x2 x

25) 2 14 x+ < 2x+ +22xx

26) 3 13 x+ < 2x+ 72x2 x

27) 3x > 1 + x1 28) 3 2(x + 1) + (x + 1) + 3x x+1 > 0

29) x 1 + x + 3 + 2 (x 1)(x + 3) > 4 2x 30) 2 2x + 1 x x. 1 x

31) x + 5 + x 3 < 1 + (x + 5)(x 3) 32) 2x 35x+ > 12x 1

33) 27x+7 + 7x6 + 2 49x +7x 42 3 5

x 4

35) 22x + x + x + 7 + 2 x + 7x 35 36)2 2

1 3x+1>

1x 1x

37) 2 2 2x 4x + 6 + x 4x + 8 2x 8x + 32 38)2

2 2

2 2

5a2(x+ x +a )

x +a

39) 2 2x 1 2x x +2x 40) 2 2x 1 2x x 2x

41) 2x1 x( x1 x ) + x x 42)2 2

1 3x+ 1 >

1 x 1 x

43) 3 3(4x 1) x +1 2x + 2x + 1 44) 22x +12x +6 2x 1 > x +2

45) x 1 + x + 3 + 2 (x 1)(x + 3) > 4 2x 46) 22x 6x + 8 x x 2

47) x + 5 + x 3 < 1 + (x + 5)(x 3) 48) 3 2 2x 2x + x x x + x 2x


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49) 27x+7 + 7x6 +2 49x +7x42 3 5

x 458)

22 2

2 2

5a2(x + x + a ) (a 0)

x + a

PHNG PHP HM S:1) x+1 + 2x+3 > 5 2) x+9 + 2x+4 > 5

3) 2x+1 > 7 x 4) 31 x < x + 5

5) 2 3x + 1 1 2x + x x 6) 2 2x 2x + 3 x 6x + 11 > 3 x x 1

7) 2x + x 1 1 8) 2x 1 + x 1 (x + 1)(3 x)

9) 2 2 2 23x 7x + 3 + x 3x + 4 > x 2 + 3x 5x 1 PHNG PHP NH GI:

(nh gi bng BT):

1) 2 2x + x 1 + x x +1 x+1 2)2x

1 + x + 1 x 2 +4

3

3) 2 2x x 1+ x + x 1 2 4) 1 + x 1 x x

5) 2 22x + 4 + 2 2 x 2 6 6) 22x 10x + 16 x 1 x 3

7) 2 4 22 x x + x + 1 x 1 + 2 8) x + 2 x 1 + x 2 x 1 2

9) 2 2 4 3 2(2x 3x + 1) 4x 20x + 25x < 2x + 1 10) 3 2 3x 2 2 x 11) x x 2 >

1 x + x 1 x x x

(nh gi bng o hm):

1) 5 5(1 x) + (1 +x) 4 2 2)2x

1 + x + 1 x 2 4

3) 3 20023x +1 + 2x +4 < 3 x189

4) 3 22x + 3x + 6x + 16 > 2 3 + 4 x

5)2 2 3 23

x + (1 x ) 27

Phu Dao Dai So 10 CA Nam

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