Question

Let a > 0 and f be continuous on [-a, a]. Suppose that f'(x) exists and...

Let a > 0 and f be continuous on [-a, a]. Suppose that f'(x) exists and f'(x)<= 1 for all x2 ㅌ (-a, a). If f(a) = a and f(-a) =-a. Show that f(0) = 0.

Hint: Consider the two cases f(0) < 0 and f(0) > 0. Use mean value theorem to prove that these are impossible cases.

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