Let f: R - R be a twice-differentiable function with f''(x) 0 for all x in R. Given that f(1/2) = 1/2 and f(1) = 1, which of the following must be true?

Question

Let f: R -> R be a twice-differentiable function with f''(x) > 0 for all x in R. Given that f(1/2) = 1/2 and f(1) = 1, which of the following must be true?

Accepted Answer

f'(1) > 1. Since f is convex (f'' > 0), its derivative f' is strictly increasing. By MVT there exists c in (1/2, 1) with f'(c) = (1 - 1/2)/(1 - 1/2) = 1. Since f' is strictly increasing and 1 > c, we have f'(1) > f'(c) = 1.

Let f: R -> R be a twice-differentiable function with f''(x) > 0 for all x in R. Given that f(1/2) = 1/2 and f(1) = 1, which of the following must be true?

Solution

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