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if a>0 ,b>0 ,c>0 and the maximum value of a(b 2 +c 2 ) +b(c 2 +a 2 ) +c(a 2 +b 2 ) is X abc ,then X is (a) 2 (b) 1 (c)6 (d) 3

4 years ago

Answers : (2)

Vikas TU
13786 Points
For Maximum value,
of  a(b2+c2) +b(c2+a2) +c(a2+b2)
every term needs to be maximum.
= > that is considering a b c be triangle sides 
with a as hypotenuse for maximum for a(b2+c2)
b hypotenuse for b(c2+a2)
c hypotenuse for c(a2+b2).
Thus from pythogorus for every term
a^2 = b^2 + c^2
simlarly for last two term.
put it in the eqn.
U would get
a^3 + b^3 + c^3 = 3abc
X = 3
from the identity.
4 years ago
mycroft holmes
272 Points
From AM-GM inequality a(b^2+c^2) \ge 2abc with equality when b=c.
Repeat this for the other two terms and add and you will get RHS \ge 6abc with equality when a=b=c
4 years ago
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