Suppose it has more than one, for example, 2. Then the mean value theorem says that there is a number in the interior of [-2, 2] where the first derivative is 0. But f'(x) = 3x^2 - 15 = 3 (x^2 - 5), and its roots are clearly outside the interval.
The other two problems are similar. The result you're using here is called Rolle's Theorem, a special case of the Mean Value Theorem.