f (x) = ax³ x² where x > 0

1. Find the value of x that maximizes or minimizes f(x) for a given value of a. Check SOC.

2. Let x(a) be the optimal x chosen for a given value of a, and let g(a) = f(x(a)) be the function representing f at the optimal x* = x(a) for a given value of a. Calculate g(a) and g'(a).

3. Now, use the envelope theorem to calculate g'(a), and confirm that you get the same answer as in 2).