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

Answer: 
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) $(a - b)^2$.
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-Schwarz applied to the vectors $\vec{x}=(1,1)$ and $\vec{y}=(a,b)$.