If $a,b \in \Bbb [0,2]$ and $a+b=2$, maximize $ab(a^2+b^2)$ without differentiation.

Question

If $a,b \in \Bbb [0,2]$ and $a+b=2$, maximize $ab(a^2+b^2)$ without differentiation.

My try

$a+b=2 \iff a=2-b \implies \max \{ab(a^2+b^2)\} \iff \max \{[(2-b)b][(2-b)+b^2)\}$

$\iff \max \{-2 b^4 + 8 b^3 - 12 b^2 + 8 b\}$

At this point, i can use differentiaton to find the maximum. It's easy to see that $\max \{-2 b^4 + 8 b^3 - 12 b^2 + 8 b\}=2$ at $b=1$ but this exercise is meant to be done without differentiation.

Any hints?

Answer 1

You can use the fact that\begin{align}ab(a^2+b^2)&=ab\bigl((a+b)^2-2ab\bigr)\\&=ab(4-2ab)\\&=2\bigl(2ab-(ab)^2\bigr)\\&=2\left(1-\left(ab-1\right)^2\right)\\&=2-2(ab-1)^2.\end{align}So, the maximum is $2$.