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Maxima, Minima and Inequalities- The Basicsby Total Gadha - Monday, 20 October 2008, 10:40 AM

This is a month of distress; students going into depression over their marks, everyone asking forattention, frantic phone calls and emails, long hours of workshops, motivating speeches, exclusivesessions. In short, bullets flying all around and everyday becoming a war zone. Part of the game though.This is the month every instructor in the field tightens his belt and gets ready for the barrage of queriesand emotions flying his way. (And I just burnt my tea I left on the burner 10 minutes ago while writingthis. Oh well!) I am still amazed how crushing those meaningless percentiles can be to the spirits of thestudents. I keep on telling students don't take your percentiles seriously. Don't take your percentilesseriously but my exhortations always fall on deaf ears. Students are so much caught in this web that they

cannot detect that half their miseries are emanating from something that is not real and cannot supplant the real thing- The CAT. Oh well, Ibetter go and answer those distress calls. For all those students telling me that I have disappeared from TG, here is the new chapter to

shush them for a while. Till the mutiny rises again



I shall have to end here and leave the rest of it for my CBT Club students. I shall cover some problems based on this inthe CBT Club this week.

If you think this article was useful, help others by sharing it with your friends!

You might also like:Quadratic EquationAbsolute Value (Modulus) Function



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Re: Maxima, Minima and Inequalities- The Basicsby vamsi krishna - Monday, 20 October 2008, 10:55 AM

Oh mY......

Inequalities Simplified....

SIR,,,we demand some exercises to munch on

VaMsI

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Re: Maxima, Minima and Inequalities- The Basicsby Kitty Witty - Monday, 20 October 2008, 06:12 PM

Sir Hats off to you for such an insightful article.

Could you plz help me solve this problem using AM > GM funda....

Let x,y,z be distinct positive integers such that x+y+z=11. Find the maximum value of (xyz+xy+yz+zx)?

This is how i approached the problem.

for product of any 2 nos to be maximum they shld be as close to each other as possible......

using this x=2,y=4,z=5

substituting (xyz+xy+yz+zx) =78

However using AM > GM funda is more fool proof....

can some one help me with that?

Regards,

Kitty Witty

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Re: Maxima, Minima and Inequalities- The Basicsby ATOM ANT - Monday, 20 October 2008, 06:52 PM

Thanks for the lesson sir..

Can you explain how you found the base and height of the triangle in the second example...

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Re: Maxima, Minima and Inequalities- The Basicsby Total Gadha - Monday, 20 October 2008, 07:14 PM

Hi Kitty,

take 4, 4, 3. The numbers should be nearly equal.

Maximum value = 88.

Total Gadha

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Re: Maxima, Minima and Inequalities- The Basicsby rashi agarwal - Monday, 20 October 2008, 07:51 PM

thankyou TG sir for such a wonderful article.We are really in need of more such articles.sir I have a doubt in this ques.

find the minimum value of |x-1|+ |x-3| +|x-10|.

In this, can we take any two values as a and b so that |x-a| +|x-b| remain constant .I havnt got the solution of this one.why have we taken aand b as 1 and 10? wht not 3 and 10?there is one more question of the same type. i havent got that also.please explain this question.

regardsrashi

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Re: Maxima, Minima and Inequalities- The Basicsby whirl wind - Tuesday, 21 October 2008, 01:28 AM

TG,

Glad to see that. If only all that u need is a mutiny for ur presence here - why - u can see one any time...and i dont mind starting one rightnow - and i dont think others will be late in joining me in the mutiny for an article of urs. U've unanimousy established the unmatched qualityof content on TG many times before...we need nt tell wat the content of the next article shd be - u know it much better. But we will be waiting for

one.

Btw, TG, wat about the solutions of CC-4 and CC-6??Havent had them yet - desperately waiting for them..

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Re: Maxima, Minima and Inequalities- The Basicsby Kitty Witty - Tuesday, 21 October 2008, 06:18 AM

Hi TG Sir,

Thanks for ur quick reply.

However the question states that x,y and z are distinct integers.

so 4,4,3 will be ruled out.

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Re: Maxima, Minima and Inequalities- The Basicsby Kitty Witty - Tuesday, 21 October 2008, 07:13 AM

Another Problem Sourced from Quant Marathon Blog

1)Two real non negative numbers satisfy that ab>=a^3+b^3, find the maximum value of a+b

a) 1/2 b) 1 c) 3/2 d) 2 e) none of these

My Approach

a^3+b^3 >= ab(a+b) if a>0,b>0 .....................1

This implies.... the least value possible of a3+b3 = ab(a+b)

a^3+b^3



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Re: Maxima, Minima and Inequalities- The Basicsby zico on the run - Tuesday, 21 October 2008, 07:26 PM

Hi Tg ,

wanted to clarify whether the " the condition for ax2 + bx + c>=0 is a>0 and b2-4ac



Re: Maxima, Minima and Inequalities- The Basicsby Chinmay Korhalkar - Wednesday, 22 October 2008, 06:56 PM

Answers :

1) 20302) 4534

Could you please elaborate on the method used? Is there some other method except making one the greatest and other two as small aspossible?

Thanks in advance,Chinmay.

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Re: Maxima, Minima and Inequalities- The Basicsby ashish sharma - Wednesday, 22 October 2008, 09:26 PM

Hi...TG Sir....in the era of commercialisation of education and exploitation of money...some very noble and generous concepts come from yourside.Sir i have joined a coaching center, paid a hefty amount and taught by many teachers but honestly the concepts you give are so grasping anduseful that i have found them much more useful than any material.I have made up my mind if in this year i dont get selected i will join yourcoaching ....

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Re: Maxima, Minima and Inequalities- The Basicsby Rishi Kapoor - Wednesday, 22 October 2008, 09:34 PM

How do we came to know which one we have to make the greatest?Please Reply

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Re: Maxima, Minima and Inequalities- The Basicsby Nikhil Dhanda - Thursday, 23 October 2008, 11:52 AM

hey my approach for this was :

given : ab >= a^3+b^3

but we know that (a^3+b^3)/2 > ((a+b)/2)^3 from theorem given above

hence

ab >= ((a+b)/2)^3

but we know that AP>=GP

ie (a+b)/2 >= (ab)^1/2

taking squares

((a+b)/2)^2 >= ab >= ((a+b)/2)^3

hence

a+b

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Re: Maxima, Minima and Inequalities- The Basicsby Nikhil Dhanda - Thursday, 23 October 2008, 03:24 PM

Whats the approach for this qs??...people who got the answer plz give ur approach for the same

Thnks

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Re: Maxima, Minima and Inequalities- The Basicsby dhwani parikh - Sunday, 9 November 2008, 09:47 PM

just superb..it helped me a lot in these last days... and the best thing is chess picture .... u've related it with this wonderfully.. great one...

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Re: Maxima, Minima and Inequalities- The Basicsby nitesh agarwal - Saturday, 15 November 2008, 03:12 PM

the other value is 1 when x=y=0 and z=1

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Re: Maxima, Minima and Inequalities- The Basics



by shivani tiwari - Thursday, 27 November 2008, 11:41 AM

Although I am not so regular on TG but whenevr I get a chance to log on I just find a single word "MIND BLOWING".

Last time I get to know how to find the last two digits of a number raised to any power ..that was yet another fantabulous article by TG so lucid..the same is with this one..

Really you are simply mindblowing in your way of explaining..

Thanks for all your efforts..hope to find some more articles on these topics..

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Re: Maxima, Minima and Inequalities- The Basicsby amit ranjan - Thursday, 30 July 2009, 07:38 PM

Hi TG,

Thanks a lot TotalGadha bhai. Very very useful stuff.

Best Regards,

Amit

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Re: Maxima, Minima and Inequalities- The Basicsby Nitin Kumar - Saturday, 5 September 2009, 01:09 PM

we need to take distinct postive integers.

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Re: Maxima, Minima and Inequalities- The Basicsby Arundeep Raina - Wednesday, 16 September 2009, 02:45 PM

Plz explain if there is any other approach to be used in such questions ( apart from making one number greatest and the others as least aspossible). can anybody explain the solution for 2nd ques?

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Re: Maxima, Minima and Inequalities- The Basicsby Ankit Talwar - Monday, 26 October 2009, 03:06 PM

Hi all,

The approach to solve the above mentioned problems is to maximize one of the numbers while minimizing the other two.

Ans 1: Take x=50, y=0,z=0; the maximum value of ( x^2 + y^2 + z^2) = 2500.

Ans 2: We have to see the weights attached with different numbers. In the second problem. z has the highest weight attached to it { 8(Wz) > 5(Wy) >2(Wx)}. So we will take the

following case x=0, y=0, z=25; the maximum value of (2x^2+ 5y^2 + 8z^2) = 5000 (which is not present as any of the options.)

Chinmay I would request you to check the answer for the second and provide clar ifications if required.

Regards

Ankit

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Re: Maxima, Minima and Inequalities- The Basicsby shravan kumar - Sunday, 8 November 2009, 04:43 PM

@ Ankit....Dude...please redo your calculations cos x, y, z are natural numbers.

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Re: Maxima, Minima and Inequalities- The Basicsby nishchai nevrekar - Sunday, 8 November 2009, 06:42 PM

ne one with viable solns to these problems.... so tht thr is some generalized method which can be xtended to other problems like these....

@chinmay. .... dude can u post those aimcat soln if possible...

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Re: Maxima, Minima and Inequalities- The Basics



by nidhi soni - Saturday, 14 November 2009, 10:20 PM

really really very gud chp tg

thnku so much

its really helpful

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Re: Maxima, Minima and Inequalities- The Basicsby payal saraf - Sunday, 29 November 2009, 05:14 PM

plz help

in d ques.(eg. 6) " find max n min value f function y= x/(x2 -5x +9)

after d step (11y+1)(y-1) less than equal to 0, we can hv 2 cases

1. 11y+1 less than equal to 0 and y-1 greater than equal to 0

therefore, y less than equal to -1/11 and y greater than equal to 1

or 2. 11y+1 greater than equal to 0 and y-1 less than equal to 0

therefore -1/11 (less than = to) y (less than = to ) 1

then why do we choose case 2 only?????

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Re: Maxima, Minima and Inequalities- The Basicsby Subhash Medhi - Tuesday, 1 June 2010, 02:39 AM

Dear TG sir, I too have the same doubt as Saurabh K. Is it possible in case of symmetrical expressions to distinguish between maximum andminimum values. In the article, it is given that to find maximum or minimum values in case of symmetrical expressions we have to assignequal values to each of the variables.Does it mean that in case of symmetrical expressions maximum and minimum values are equal ? Canthat really be the case ? Or should we assign equal values to the variables only while calculating the minimum value?

Regards,Subhash

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Re: Maxima, Minima and Inequalities- The Basicsby Pravin Vaidya - Wednesday, 2 June 2010, 12:36 AM

simple doubt??

solve the inequality...(X/4) + (2/3) < (2X/3)-(1/6)==>

1st approach

(X/4)-(2X/3) < -(1/6) -(2/3)-5X/12 < -(5/6)===> (X/12) < (1/6)

which gives,

X



See -5 < -2

does that mean 5 < 2.

I hope you can find your error now.

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Re: Maxima, Minima and Inequalities- The Basicsby Pravin Vaidya - Thursday, 3 June 2010, 01:39 PM

Thanks kamal....

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Re: Maxima, Minima and Inequalities- The Basicsby Naman Mirchandani - Saturday, 19 June 2010, 11:25 AM

Sir,

For the 2nd last question where a+b+c = 1,

how will we know that putting a=b=c, will give the mxm or minm value of

(1/a - 1) (1/b - 1) (1/c - 1) ?

Thanks,

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Re: Maxima, Minima and Inequalities- The Basicsby Ramakanth Kanagovi - Monday, 12 July 2010, 04:30 PM

1)Two real non negative numbers satisfy that ab>=a^3+b^3, find the maximum value of a+b

a) 1/2 b) 1 c) 3/2 d) 2 e) none of these

i think the answer is 1/2 taking the values as 1/4 and 1/4

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Re: Maxima, Minima and Inequalities- The Basicsby Abhirup DebRay - Monday, 12 July 2010, 07:34 PM

thnk option b 4 a=b=1/2

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Re: Maxima, Minima and Inequalities- The Basicsby Navneet H - Wednesday, 22 December 2010, 07:45 PM

Hi kitty,

The values that you have taken are not close to each other.The value of x y and z should be 11/3.. in that case the result of te expression willbe: 2429/27 which is slightly greater than 89.666

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Re: Maxima, Minima and Inequalities- The Basicsby rajesh mishra - Saturday, 30 July 2011, 01:00 AM

what in case if the mod value changes like |1-x|+|2-x|+|3.5-x|+|x-4| can u please explain this in terms of distances as explained above?

thanks

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Re: Maxima, Minima and Inequalities- The Basicsby manisha dalan - Tuesday, 2 August 2011, 04:55 PM

dear sir,



I am really grateful to you for this wonderful article.

Regards,

manisha.

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Re: Maxima, Minima and Inequalities- The Basicsby TG Team - Wednesday, 3 August 2011, 01:46 PM

Hi Rajesh

Understand the concept clearly. On number line '5' denotes a point which is 5 unit away from origin on the right side.Right?

Similarly 'x' denotes a point which lies at a distance of 'x' units from origin. So |x - 5| denotes the distance between twopoints 'x' and '5' on the number line.

What does |5 - x| represent on number line?

Isn't it same the distance between two points '5' and 'x' on the number line? It is.

Hope it is clear.

Kamal Lohia

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Re: Maxima, Minima and Inequalities- The Basicsby anupam chaturvedi - Thursday, 25 August 2011, 03:33 PM

Hi TG Sir,

I am still not able to understand the concept of symmetrical expressions. What does it exactly mean for below questions!

1) --> Min of (a1+a2+a3+a4)(1/a1 + 1/a2 + 1/a3 + 1/a4)I solved it using max product rule.How to apply symmetry here?

--> a+b+c=18 and we need to find MIN of (1/a -1)(1/b -1)(1/c -1)Is (a+b+c) called symmetrical OR (1/a -1)(1/b -1)(1/c -1) ?

2) (a1+a2+a3+a4)(1/a1 + 1/a2 + 1/a3 + 1/a4)how can we find the max. value of above function..??

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Re: Maxima, Minima and Inequalities- The Basicsby abhinay dutta - Thursday, 8 September 2011, 11:26 AM

Sir Hats off to you for such an insightful article.

Could you plz help me solve this problem using AM > GM funda....

Let x,y,z be distinct positive integers such that x+y+z=11. Find the maximum value of (xyz+xy+yz+zx)?

This is how i approached the problem.

for product of any 2 nos to be maximum they shld be as close to each other as possible......

using this x=2,y=4,z=5

substituting (xyz+xy+yz+zx) =78

had u calculated values for x,y,z as 4,4,3 answer wud hv been 88

more than 78 and is Maximum.

Guys can u post more question on Max and min or can u site some sources for practise.

Thanks

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Re: Maxima, Minima and Inequalities- The Basicsby TG Team - Thursday, 8 September 2011, 12:03 PM

Hi Abhinay

xyz + xy + yz + zx = (x + 1)(y + 1)(z + 1) - (x + y + z) - 1

And this will be maximum when (x + 1)(y + 1)(z + 1) is maximum.

We know that sum of these three terms (i.e. x + 1 + y + 1 + z + 1 = 14) is constant, so there product will be maximumwhen these three terms are as close as possible.(preferably equal)

But in this question, it is given that x, y, z are distinct positive integers so (x + 1), (y + 1), (z + 1) should also bedistinct but should be close also. So the optimum case is 6, 5, 3 and the required maximum product will be 6 5 3 -11 - 1 = 78.

Kamal Lohia

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Re: Maxima, Minima and Inequalities- The Basicsby priyanka j - Thursday, 8 September 2011, 02:08 PM

f(x)=min(5-x,x+3), find max value of f(x)

Sir, In these type of questions is it necessary to draw graph can't we directly find out the intersection of both lines by directly putting them equalto each other.

like in this case 5-x=x+3 gives x=1 . max value of f(x)=4 at x=1

Thanks

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Re: Maxima, Minima and Inequalities- The Basicsby TG Team - Thursday, 8 September 2011, 02:38 PM

Priyanka

In this case, it's ok to just equate the expressions and get the value of x at which f(x) attains its maximum.

But if f(x) = min(5 + 3x, x + 3), then what will you do?

Just try and think.

Kamal Lohia

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Re: Maxima, Minima and Inequalities- The Basicsby priyanka j - Thursday, 8 September 2011, 03:25 PM

Thanks sir.it will nt work in evry case. but in d ques u have written is it possible to find any mix max or min value. bcz d value ranges frm -infinity to +infinity. As value of f(x) isincreasing as x increases & decreasing as x decreases. What i think is there must be any limit for value of x, then only it is possible to find minor max value of f(x).

I know m asking silly things but plz clear my doubts.

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Re: Maxima, Minima and Inequalities- The Basicsby Jitendra Soni - Friday, 9 September 2011, 03:18 PM

My solutions:A) using x^2 + y^2 + z^2 >= ((x+y+z)^2)/3 ..... (i) , we get from the given condition 2x+3y+4z=1002(x+y+z) + y+2z=100 or x+y+z= 50-(y+2z)/2 ..now y + 2z should be divisible by 2. As all x,y,z are natural nos. hence for max value of x + y +z, we need min value of y+2z (which should be even also to give integral value of x+y+z.Trying with y+2z =4 , we get y=2 and z=1 (with 2 wewill not get natural nos.) hence x+y+z = 48. Now substitute in x^2 + y^2 + z^2 to get the ans B) separate all terms, we get x^2+ x^2+ y^2+ y^2+ y^2+ y^2+ y^2+ z^2+ z^2+ z^2 +z^2+ z^2+ z^2+ z^2+ z^2 >=((x+x+y+y+y+y+y+z+z+z+z+z+z+z+z)^2)/15 = ((2x+5y+8z)^2)/15..= ((200-(2x+y))^2)/15 , now we have to minimize 2x+y subject to thecondition that z = ( 25-(2x+3y)/4) will be an integer. This gives 2x+3y=8 ( = 4 does not give natural no. sol.) hence x= 1, y= 2 and z= 23.Substitution in 2x^2+5y^2+8z^2 gives the ans.

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Re: Maxima, Minima and Inequalities- The Basicsby sandesh gupta - Monday, 12 September 2011, 04:47 PM



Thanks a lot sir for such a wonderful article . Sir can you please explain how yo solve question like |X -3| + |Y - 4| = 5 (for ex) kind of question. I am always stuck in this kind of problem .

Thanks & Regards

sandesh(coming out from my IT background )

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Re: Maxima, Minima and Inequalities- The Basicsby Rahul Sharma - Wednesday, 14 September 2011, 11:33 AM

Hi sir,

min value of |x-1|+|x-10|

as per my understanding, the value of the expression will be a constant when x lies between max and the min values(i.e 1, 10). when there areother terms included in between then the value of the expression will largely depend on them.if we have odd terms in all, then the value of theexpression will b max at the value of x obtained from the middle term. If the no of terms are even the the two middle terms will decide the range of x for the expression tohave max value.

please let me know if i m right. And alsoe please help me in finding the values of the following expressions:

|1-x|+|x-3|+|x-10||x^2-1|+|x2-3|+|x^2-10|

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Inequalities conceptsby Dinesh H - Thursday, 21 February 2013, 12:00 AM

For Inequalities concepts pls visit http://start-from-scratch-cat.blogspot.in/2013/02/cat-inequalities-concepts.html

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Re: Maxima, Minima and Inequalities- The Basicsby ashwini rathore - Friday, 22 February 2013, 11:41 PM

hi this is yash...i m going to tell my methodF(X)=|X-a|+|X-b|+|X-c|min value always occur At b if a7.5

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