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.

To determine which statement must be true given the conditions, let's analyze each option: 1. **Given Conditions:** - \( f \) is continuous on the closed interval \([-2, 5]\). - \( f \) is differentiable on the open interval \((-2, 5)\). - \( f(-2) = f(5) = 3 \). 2. **Applying Rolle's Theorem:** Since \( f \) is continuous on \([-2, 5]\) and differentiable on \((-2, 5)\) with \( f(-2) = f(5) \), Rolle's Theorem guarantees that there exists at least one number \( c \) in \((-2, 5)\) where \( f'(c) = 0 \). 3. **Evaluating the Statements:** - **(A) There is a number \( c \) in the interval \((-2, 5)\) for which \( f(c) = 0 \).** - **Not necessarily true.** The function could maintain a constant value of 3 without ever reaching 0. - **(B) \( f'(x) > 0 \) for all numbers in the interval \((-2, 5)\).** - **Not true.** By Rolle's Theorem, there exists at least one \( c \) where \( f'(c) = 0 \), which contradicts the statement. - **(C) \( f'(x) = 3 \) for all numbers in the interval \((-2, 5)\).** - **Not necessarily true.** The derivative could vary and is only required to equal zero at some point by Rolle's Theorem. 4. **Conclusion:** Since none of the statements (A), (B), or (C) must be true based on the given conditions, the correct answer is: **Answer:** (D) None of the above