Let f: R - R be a twice differentiable function such that 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 about f'(1)?

Question

Let f: R -> R be a twice differentiable function such that 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 about f'(1)?

Accepted Answer

f'(1) > 1/2. Since f'' > 0, f is strictly convex. The slope of the chord joining (1/2, 1/2) to (1, 1) is (1 - 1/2)/(1 - 1/2) = 1. For a strictly convex function, f'(1) >= slope of chord = 1. Since 1 > 1/2, we have f'(1) > 1/2. In fact f'(1) >= 1 > 1/2, confirming option D.

Let f: R -> R be a twice differentiable function such that 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 about f'(1)?

Solution

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