Let f: R → R be a function that is twice differentiable, satisfying f''(x) > 0 for every x in R, with f(1/2) = 1/2 and f(1) = 1. Which of the following is true about f'(1)?

1/2 < f'(1) ≤ 1
f'(1) is greater than 1
0 < f'(1) ≤ 1/2
f'(1) is less than or equal to 0

Correct answer: f'(1) is greater than 1

Solution

Given that f''(x) > 0 for every x in R, the function f'(x) is increasing, and using the initial conditions f(1/2) = 1/2 and f(1) = 1, we can conclude that f'(1) is greater than 1.

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