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Inequalities

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 (a+b)(b+c)(c+a)>=8abc

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 x^4+y^4+z^4 =27 find min value of x^6+y^6+z^6

use cauchy on x^3,y^3,z^3 and x,y,z
then (x^6+y^6+z^6)(x^2+y^2+z^2)>=(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 (x^4+y^4+z^4)(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)<sup>4<=3[(x^6+y^6+z^6)</sup>2](x^4+y^4+z^4)
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: Find the min value of 2x+y if xy=8 and x,y are positive numbers

shivam01
  • Authority 27
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shivam01 said:

helps me in getting familiar to CAUCHY”S

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  • Posted about 1 month ago.
Riyana
  • Authority 124
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Riyana said:

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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  • Posted about 1 month ago.
Sureshbala
  • Authority 700
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Sureshbala said:

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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  • Posted about 1 month ago.
asureshwaran
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asureshwaran said:

re sureshbala this lesson is pretty good for cat aspirants but useless for gre test takers

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  • Posted 2 days ago.
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