Online CAT 2009 - Quant Section - Inequalities

Person

Sureshbala

Online CAT 2009 - Quant Section - Inequalities

Author: Sureshbala

%{font-family:verdana}Inequalities are generally present in CAT and similar MBA papers, the question can be direct or indirect.

AM-GM Inequality

It means that AM( arithemetic mean) of a set of positive numbers is always greater than or equal to the GM( geometric mean). The equality holds when the numbers are equal
$\frac{(a+b+c)}{3} >$ $=(a+b+c)^{(1/3)}\cdots \cdots (2.1)$
Example 1: If a,b,c are positive numbers prove that

what we will do is use AM-GM multiple times

$\frac {(a+b)}{2}> = \sqrt{(ab)}$
$=>(a+b)>=2\sqrt{(ab)}$

similarly for others

$(b+c) >=2\sqrt{(bc)}$
$(c+a) >=2\sqrt{(ac)}$

then multiplying these three inequalities we get the desired result!

Practice Problem 1: show that $(n^n)[(n+1)/2]^{(2n)}>(n!)^3$

Practice Problem 2: if x,y,z be the lengths of the sides of a triangle then prove that (x+y+z)^3>=27(x+y-z)(y+z-x)(z+x-y)
Practice Problem 3: show that for any natural number $n,(n+1)^n >2.4.6\cdots 2n$

Example 2: Show that for any natural number $n 2^n >=1 +n.2^{[(n-1)/2]}$

Lets see how we do this

$2^n>=1+n.2^{[(n-1)/2]}$
$2^n-1 >=n.2^{[(n-1)/2]}$ ( can you recognise the form?)

its the sum of a GP
we need to use AM-GM on the sum of GP

$[1+2+2^2\cdots +2^{(n-1)}]/n>(1.2.2^2\cdots 2^{(n-1)})^{(1/n)}$
$(2^{n-1})/n >$ $(2^{(1+2+3\cdots +n-1)})^{(1/n)}$ $=(2^{[n(n-1)/2]})^{(1/n)}=2^{((n-1)/2)}$

so

$2^{n-1}>2^{((n-1)/2)}$

so we are done !!

Cauchy- Schwartz Inequality

If a,b,c and x,y,z be real numbers ( positive, negative or zero) then
(ax+by+cz)^2<=(a^2+b^2+c^2)(x^2+y^2+z^2)
Equality holds iff a:b:c::x:y:z

Example 3: if find min value of

use cauchy on x^3,y^3,z^3 and x,y,z
then (x^6+y^6+z^6) $>=(x^4+y^4+z^4)^2\cdots (1)$
use cauchy on the numbers x^2,y^2,z^2 and 1,1,1
then (1+1+1)>=(x^2 + y^2 + z^2)^2
3(x^4+y^4+z^4) >=(x^2+y^2+z^2) $^2\cdots (2)$
squaring both sides of 1 and using 2 we get
(x^4+y^4+z^4) ^4 <= 3[(x^6+y^6+z^6) ^2]
putting x^4+y^4+z^4=27 and taking positive square root we get
x^6+y^6+z^6 >=81

Practice Problem 4: if a,b,c be positive numbers such that a+b+c=4 find minimum value of a^3+b^3+c^3

Practice Problem 5:

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11 Comments

shivam01 said – Thu, 13 Nov 2008 04:33:37 -0000 ( Flag Edit Link )

helps me in getting familiar to CAUCHY”S
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Riyana said – Sun, 23 Nov 2008 10:08:12 -0000 ( Flag Edit Link )

If u could provide some more information, be more precise then it wud be much more helpful. Well thank u very much for ur contribution.
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Sureshbala said – Thu, 27 Nov 2008 07:05:13 -0000 ( Flag Edit Link )

Dear Riyana, this is a short lesson to revise some of the important concepts of Inequalities. In the very near future we will come up with lessons, exclusively on Inequalities, that will focus right from basics to advanced concepts.
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asureshwaran said – Sun, 04 Jan 2009 04:11:01 -0000 ( Flag Edit Link )

re sureshbala this lesson is pretty good for cat aspirants but useless for gre test takers
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1. chandra_avinash said – Sun, 28 Jun 2009 18:13:51 -0000 ( Flag Edit Link )
  
  I disagree.
  
  It’s possible that you encounter such questions in the GRE too. You might end up getting a couple of tough questions at the end – and it’s possible that one of these theorems save the day for you!
  
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catxat said – Tue, 13 Jan 2009 17:40:19 -0000 ( Flag Edit Link )

its really good but more explanation needed
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Sureshbala said – Fri, 16 Jan 2009 17:46:36 -0000 ( Flag Edit Link )

Dear catxat, very soon we are going to come up with a lesson on Inequalities where all the concepts right from the basics will be discussed.This lesson is written keeping in mind the CAT/XAT/IIFT test takers of last year
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sankarapandian said – Fri, 13 Mar 2009 19:49:27 -0000 ( Flag Edit Link )

very gud sir.make these type of articles more sir
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userdce said – Sat, 30 May 2009 15:29:58 -0000 ( Flag Edit Link )

gud
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userdce said – Sat, 30 May 2009 15:51:57 -0000 ( Flag Edit Link )

give the answers of problems at least
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bhamshu said – Mon, 01 Jun 2009 03:33:59 -0000 ( Flag Edit Link )

isn’t the 3rd line wrong???? i yhink it should be…. (a+b+c)/3 >(abc)^(1/3)
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1. Sureshbala said – Thu, 04 Jun 2009 10:23:24 -0000 ( Flag Edit Link )
  
  No…that’s absolutely correct. The equality holds when the numbers are equal
  
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2. chandra_avinash said – Sun, 28 Jun 2009 18:16:53 -0000 ( Flag Edit Link )
  
  The 3rd line is correct.
  
  Assume a, b and c to be equal to 1; that’s when (a+b+c)/3 = 1 and (abc)^(1/3) is also = 1
  
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nehassweet1 said – Sun, 07 Jun 2009 11:37:19 -0000 ( Flag Edit Link )

the lesson is fabulous but please provide the answers to practice problems
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1. chandra_avinash said – Sun, 28 Jun 2009 18:17:45 -0000 ( Flag Edit Link )
  
  the practice problems are meant for you to work out on your own;
  
  where are you getting stuck at? just let us know and we can give you a couple of hints or point you in the right direction;
  
  keep trying…that’s the best way to crack the problem :)
  
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