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 f ′(x2) is negative.
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