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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 for a ttention, frantic phone callsand emails, long hours of workshops, motivating speeches, exclusive sessions. In short, bullets flying all around and everydaybecoming a war zone. Par t of the game though. This is the month every instructor in the field tightens his belt and gets readyfor the barrage of queries and emotions flying his way. (And I just burnt my tea I le ft on the burner 10 minutes ago whilewriting this. Oh well!) I am still amazed how crushing those meaningless percentiles can be to the spirits of the students. Ikeep on telling students don't take your percentiles seriously. Don't take your percentiles seriously but my exhortations alwaysfall on deaf ea rs. Students are so much caught in this web that they cannot detect that half their miseries are emanating fromsomething that is not real and cannot supplant the real thing- The CAT. Oh well, I better 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 aga in

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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 in the CBT Club this

week.

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

You might also like:

Quadratic Equation

Absolute Value (Modulus) Function

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Re: Maxima, Minima and Inequalities- The Basics

by vamsi krishna - Monday, 20 October 2008, 10:55 AM

Oh mY......

Inequa lities Simplified....

SIR,,,we dem and som e exe rcises 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 m e 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 m aximum 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 fo ol proof....

can some one he lp me with that?

Regards,

Kitty Witty

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by ATOM ANT - Monday, 20 October 2008, 06:52 PM

Tha nks fo r the less on sir..

Can you ex plain how you found the base and h eight of the triangle in the second examp le...

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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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by 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 a and b a s 1 and 10 ?wht not 3 and 10?there is one mo re question of the sam e type. i havent got that also.please ex plain 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 o ne any time...and i dont mind starting one right now - and i dontthink others will be late in joining me in the mutiny for an article of urs. U've unanimous y established the unma tched qua lity of content on TG man y times

befo re...we ne ed n t tell wat the conte nt of the nex t article shd be - u kno w it much be tter. But we will be waiting for on e.

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

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by 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 P roblem 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 po ssible of a3+b3 = ab(a+b)

a^3+b^3


6/12

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by zico on the run - Tuesday, 21 October 2008, 07:26 P M

Hi Tg ,

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


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Hi...TG Sir....in the era o f comme rcialisation o f education and exploitation of m oney...some very noble and ge nerous concepts come from your side.Sir i have joined a coaching center, paid a hefty amou nt and taug ht by many tea chers but ho nestly the concepts you give are so grasping a nd useful that i have found

them m uch more useful than any ma terial.I have m ade u p my m ind if in this year i dont get selected i will join your coaching ....

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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 thegreatest?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 givenabove

hence

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

but we kno w 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 go t 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 P M

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.. greatone...

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Re: Maxima, Minima and Inequalities- The Basicsby nitesh agarwal - Saturday, 15 November 200 8, 03:12 P M

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

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by shivani tiwari - Thursday, 27 Novembe r 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 thisone..

Really you a re simply mindblowing in your way of explaining..

Thanks for all your efforts..hope to find some more a rticles 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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by Nitin Kumar - Saturday, 5 Septem ber 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 Septembe r 2009, 02:45 PM

Plz explain if there is any other approach to be used in such questions ( apart from m aking one numbe r greatest and the others as least as possible). can anybodyexplain 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; themaximum 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 clarifications if required.

Regards

Ankit

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

@ Ankit..

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

numbers.Show parent | Reply


by nishchai nevrekar - Sunday, 8 November 2009, 06:42 P M

ne on e with viable solns to these problems.... so tht thr is som e gene ralized m ethod 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 Basicsby nidhi soni - Saturday, 14 November 200 9, 10:20 P M

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 e qual to -1/11 an d y greater than equal to 1

or 2. 11y+1 greater than e qual to 0 a nd y-1 less than eq ual to 0

therefo re -1/11 (less tha n = to) y (less than = to ) 1

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

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by Subhash Medhi - Tuesday, 1 June 20 10, 02:39 AM

Dea r TG sir,I too have the sam e doubt as Saurabh K. Is it possible in case of symmetrical expressions to distinguish between m aximum and m inimum values. In

the article, it is given that to find maximum or minimum values in case of symm etrical expressions we have to assign equa l values to each of the variables.Does itmea n that in case of sym metrical expressions m aximum and m inimum values are eq ual ? Can that really be the case ? O r should we assign eq ual 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...

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(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=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 a nd 1/4

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Re: Maxima, Minima and Inequalities- The Basicsby Abhirup DebRa y - 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 Decembe r 2010, 07:45 PM

Hi kitty,

The values that you ha ve taken a re not close to each other.The value of x y and z sho uld be 11/3.. in that case the result of te exp ression will be: 2429/27 which is

slightly greater than 89.666

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by rajesh mishra - Saturday, 30 July 2011, 01:00 AM

what in case if the mo d value changes like |1-x|+|2-x|+|3.5-x|+|x-4| can u please e xplain this in terms of distances as ex plained

above?

thanks

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

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 two points '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 P M

Hi TG Sir,

I am still not able to un derstand the concept of symmetrical expressions. W hat does it exactly mean for belowquestions!

1)--> Min of (a1+a2+a3+a4)(1/a1 + 1/a2 + 1/a3 + 1/a4)

I solved it using ma x 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 m ax. value o f above function..??

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by abhinay dutta - Thursday, 8 September 2011, 11 :26 AM

Sir Hats off to you for such an insightful article.

Could you plz help m e 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 m aximum 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 fo r x,y,z as 4,4,3 ans wer wud hv been 88

more than 78 and is Maximum.

Guys can u po st more question on Ma x and m in or can u site som e sources for practise.

Thanks

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

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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 maximum when these three

terms are as c lose 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 be distinct 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, 0 2:08 PM

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

Sir, In these type of qu estions is it necessary to draw graph can't we directly find ou t the intersection of bo th lines by directly putting them equ al to ea chother.

like in this case 5-x=x+3 gives x=1 . m ax value o f f(x)=4 at x=1

Thanks

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Re: Maxima, Minima and Inequalities- The Basicsby TG Team - Thursday, 8 Septembe r 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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by priyanka j - Thursday, 8 September 2011, 0 3:25 PM

Tha nks s ir.it will nt work in evry case.

but in d q ues u have written is it po ssible to find any m ix ma x or m in value. bcz d value ranges frm -infinity to +infinity. As value of f(x) is increasing a s x increases &decreasing as x de creases. W hat i think is there must be any limit for value of x , then only it is possible to find min or ma x value of f(x).

I kn ow 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, 0 3:18 PM

My solutions:A) using x^2 + y^2 + z^2 >= ((x+y+z)^2)/3 ..... (i) , we ge t from the given condition 2x +3y+4z=100

2(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 valueof 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 we will not get natural nos.) hence x+y+z = 48.Now substitute in x^2 + y^2 + z^2 to get the ans

B) sepa rate all te rms, 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 the condition that z = ( 25-

(2x+3y)/4) will be an intege r. This gives 2x+3y=8 ( = 4 does no t give natural no. sol.) hen ce x= 1, y= 2 and z= 23. Subs titution 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 ho w yo solve qu estion like |X -3| + |Y - 4 | = 5 (for ex ) kind o f question . I am always stuck inthis kind of problem .

Thanks & Regards

sandesh(coming out from my IT background )

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by Rahul Sharma - Wednesday, 14 September 2011, 11:33 AM

Hi s ir,

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

as pe r my understanding, the value of the ex pression will be a constant when x lies between m ax a nd the m in values(i.e 1, 10). when there are other terms includedin between then the value of the e xpression will largely depend o n them.if we have o dd terms in all, then the value of the expression will b ma x at the

value of x obtained from the middle term. If the no of terms a re even the the two middle terms will decide the range of x for the expression to have ma x value.

please let me know if i m right. And alsoe p lease help m e in finding the values of thefollowing e xpressions:

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

http://www.web2pdfconvert.com/?ref=PDFhttp://www.web2pdfconvert.com/?ref=PDFhttp://totalgadha.com/user/view.php?id=131710&course=1http://totalgadha.com/user/view.php?id=131710&course=1http://totalgadha.com/mod/forum/post.php?reply=82605http://totalgadha.com/user/view.php?id=112539&course=1http://totalgadha.com/user/view.php?id=112539&course=1http://totalgadha.com/mod/forum/post.php?reply=82551http://totalgadha.com/user/view.php?id=115580&course=1http://totalgadha.com/user/view.php?id=115580&course=1http://totalgadha.com/mod/forum/post.php?reply=82538http://totalgadha.com/user/view.php?id=93304&course=1http://totalgadha.com/user/view.php?id=93304&course=1http://totalgadha.com/mod/forum/post.php?reply=82537http://totalgadha.com/user/view.php?id=2862&course=1http://totalgadha.com/user/view.php?id=2862&course=1http://totalgadha.com/mod/forum/post.php?reply=82536http://totalgadha.com/user/view.php?id=93304&course=1http://totalgadha.com/user/view.php?id=93304&course=1http://totalgadha.com/mod/forum/post.php?reply=82533


12/12

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Inequalities concepts

by 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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