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

Answer: 
If $a+b = 23$ then $b=23-a$. So just maximize and minimize $2a^2-46a+23^2$ (for $0\le a \le 23$) using calculus.
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 $\vec{y}=(a,b)$.