Part a)

Solution: “the difference between its square and its cube” means x^2− x^3. “the greatest” implies we’re looking for a max (or at least an extremum). “between 0 and 1” means we’re on the interval [0, 1]. So

f(x) = x^2− x^3 implies f '(x) = 2x−3x^2;

this is 0 if x = 0 or 2 = 3x

meaningx = 2/3

Our values are:

f(0) = 0, f(1) = 0, f(2/3) = 4/9 - 8/27 = 4/27