Max

If a+2b+c=4 .... a,b,c Î¾ R

then max(ab+bc+ca)=?

3 Answers

by symmetry we can say that max min will be when a=c

so we can say that a=c=t and b=2-t

so we have to maximize

t(2-t).2+t² = 4t-t² =

for t in real

which is 4 :)

[1]

341

Hari Shankar ·2009-02-11 23:26:55

We are given (a+b)+(b+c) = 4.

from which we get since (x+y)² â‰¥ 4xy, 4â‰¥(a+b) (b+c)

Now (a+b)(b+c) = b²+ab+bc+ca

Hence we get ab+bc+ca â‰¤ 4-b²

Notice that the maximum of (a+b)(b+c) is attained when a=c no matter what the value of b is.

So we can have b = 0.

which means the maximum value of ab+bc+ca is 4

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