Help me answer this optimization problem.

Question

What two nonnegative real numbers $a$ and $b$ whose sum is 23 maximize $a^2+b^2$? Minimize $a^2+b^2$?

Answer 1

If $a+b = 23$ then $b=a-23$ so just maximize and minimize it $2a^2-46a+23^2$ using calculus.

Answer 2

Hint:

$$2(a^2 + b^2) = (a + b)^2 + (a - b)^2 = 23^2 + (a - b)^2.$$

So to maximize (minimize) $a^2 + b^2$, you should maximize (minimize) $(a - b)^2$.

Answer 3

You could make use of the following inequality. $$(a+b)^2 \leq 2(a^2+b^2)$$ The inequality comes from the fact that $(a-b)^2 \geq 0$ or can be thought as Cauchy-Schwartz applied to the vectors $\vec{x} = (1,1)$ and $\vec{y} = (a,b)$.