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

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 2

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.

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

Answer 4

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