Local Max & Min

Any critical points will occur when $f'(x) = -3x^{2} + 3a = 0$ , i.e., when $a = x^{2}$ . So $a \gt 0$ , and the critical points will occur when $x = -\sqrt{a}$ and $x = \sqrt{a}$ .

Now since the leading coefficient of this cubic polynomial is negative we know that the minimum will be $f(-\sqrt{a})$ and the maximum will be $f(\sqrt{a})$ . So we have that

$f\sqrt{-a} = a\sqrt{a} - 3a\sqrt{a} + b = 0 \longrightarrow -2a\sqrt{a} + b = 0$ and that

$f\sqrt{a} = -a\sqrt{a} + 3a\sqrt{a} + b = 20 \longrightarrow 2a\sqrt{a} + b = 20$ .

Solving simultaneously, we have that

$4a\sqrt{a} = 20 \longrightarrow a\sqrt{a} = 5$ , and so $b = 20 - 2a\sqrt{a} = 20 - 2*5 = 10$ .

Thus $a\sqrt{a} + b = 5 + 10 = \boxed{15}$ .