1. A function \( f \) is continuous on the closed interval \( [-2,5] \), differentiable on the open interval \( (-2,5) \), and \( f(-2)=f(5)=3 \). Which of the following statements must be true? (A) There is a number \( c \) in the interval \( (-2,5) \) for which \( f(c)=0 \). (B) \( f^{\prime}(x)>0 \) for all numbers in the interval \( (-2,5) \). (C) \( f^{\prime}(x)=3 \) for all numbers in the interval \( (-2,5) \). (D) None of the above The graph of \( f^{\prime} \) for the interval \( [-3,5] \) is shown below.

The function \( f \) is supported by the fact that it is continuous and returns to the same value at the endpoints: \( f(-2) = f(5) = 3 \). This suggests that the function must oscillate at some point in the interval \( (-2, 5) \). Specifically, the Mean Value Theorem tells us that since \( f(-2) = f(5) \), there exists at least one point \( c \) in \( (-2, 5) \) where \( f'(c) = 0 \). This means there must be a maximum or minimum within that interval. For option (D), it's actually correct to choose "None of the above" because none of the other options hold true based on the given conditions: (A) does not necessarily have to be valid, and (B) & (C) imply strict behaviors for \( f' \) which are not guaranteed just because \( f(-2) = f(5) \).