Let f be a function continuous on [0, 1] and twice differentiable on (0, 1). (a) Suppose that f(0) = f(1) = 0 and f(c) >0 for some c € (0, 1). Prove that there exists xo € (0, 1) such that f"(x) < 0. (b) Suppose that ["* f(x) dx = f(0) = f(1) = 0. Prove that there exists a number xo € (0, 1) such that f"(x) = 0.

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