1) For what number between 0 and 1 is the difference between its square and its cube greatest?
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
meaning x = 2/3
Our values are:
f(0) = 0, f(1) = 0, f(2/3) = 4/9 - 8/27 = 4/27