Let $f$ be a function continuous on the closed interval $[-2, 5]$ and differentiable on the open interval $(-2, 5)$. If $f(-2) = 1$ and $f(5) = 22$, which of the following statements must be true?

Core Concept

Correct. By the Mean Value Theorem, since $f$ is continuous on $[-2, 5]$ and differentiable on $(-2, 5)$, there must exist a $c$ in $(-2, 5)$ such that $f'(c) = \frac{f(5) - f(-2)}{5 - (-2)} = \frac{22 - 1}{7} = 3$.