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

Answer: 
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 $\vec{y}=(a,b)$.
Hint:
$$a^2+b^2=(a+b)^2-2ab=23^2-2ab$$Now, it is enough to find the maximum and minimum value of $ab$.