Suppose f is continuous on [1, 5] and differentiable on (1, 5). If f(1) = 2, f(5) = 10, and f'(c) = 4 for some c in (1, 5), which of the following could be true?
Correct: B
By the Mean Value Theorem, there exists some c in (1, 5) such that f'(c) = (f(5) - f(1)) / (5 - 1) = (10 - 2) / 4 = 8 / 4 = 2. The MVT guarantees at least one point where the derivative equals 2. However, it does not forbid the derivative from taking other values. The fact that f'(c) = 4 for some c is perfectly consistent — the derivative can be 4 at some points and 2 at others. The statement 'f'(c) = 4 for some c' does not contradict the MVT; the MVT only guarantees the existence of a point where f'(c) = 2. Therefore choice B is correct: there is no contradiction; f'(c) = 4 is possible.