Question

Let a, b, c be real number such that a+2b+c=4. Find max (ab+bc+ca)

Answer 1

Consider the problem

Given,

$a,b,c$ are real number

And also, $a+2b+c=4$

$a+c=4-2b...........\left(1\right)$

$Max(ab+bc+ca)$

$\begin{array}{c}=max\left(b\left(a+c\right)+ca\right)\\ =max\left(b\left(4-2b\right)+ca\right)\\ =max\left(4b-2b^2+ca\right)\end{array}$

This is maximum when $ac>0$ and their sum is constant such that

$a=c=\left(\frac{4-2b}{2}\right)=2-b$

$\begin{array}{c}Max\left(4b-2b^2+{\left(2-b\right)}^2\right)\\ =max\left(4b-2b^2+4-4b+b^2\right)\\ =max\left(4-b^2\right)\\ =4\left[\because b^2>0\right]\end{array}$