Suppose that f is differentiable on (a, b), continuous on [a, b], and that f(a) = f(b) = 0. Prove that if f(c) ≠ 0 for some c ∈ (a, b), then there exist x1, x2 ∈ (a, b) such that f'(x1) is positive and fʹ(x2) is negative.