Given \( h(x)=f^{\prime}(x) \) where \( f \) is a function defined for \( -2 \leq x \leq 4 \) 1561 1562 Explain why \( f \) has a local mimmum to the graph of \( f \)

To determine why the function \( f \) has a local minimum based on the information given, we can analyze the relationship between the function \( f \) and its derivative \( h(x) = f'(x) \). ### Step 1: Understanding the Derivative The derivative \( f'(x) \) provides information about the slope of the function \( f(x) \): - If \( f'(x) > 0 \), the function \( f(x) \) is increasing at that point. - If \( f'(x) < 0 \), the function \( f(x) \) is decreasing at that point. - If \( f'(x) = 0 \), the function \( f(x) \) may have a local maximum, local minimum, or a saddle point. ### Step 2: Conditions for a Local Minimum A function \( f \) has a local minimum at a point \( c \) if: 1. \( f'(c) = 0 \) (the derivative is zero at that point). 2. \( f'(x) \) changes from negative to positive as \( x \) passes through \( c \) (the slope changes from decreasing to increasing). ### Step 3: Analyzing the Given Information Since we know that \( h(x) = f'(x) \), we need to find a point \( c \) in the interval \( -2 \leq x \leq 4 \) where: - \( h(c) = 0 \) - \( h(x) < 0 \) for \( x < c \) (indicating that \( f \) is decreasing before \( c \)) - \( h(x) > 0 \) for \( x > c \) (indicating that \( f \) is increasing after \( c \)) ### Conclusion If such a point \( c \) exists within the interval \( -2 \leq x \leq 4 \), then \( f \) has a local minimum at \( c \). To summarize, \( f \) has a local minimum if: - There is a point \( c \) where \( f'(c) = 0 \). - The derivative changes from negative to positive at that point. If you have specific values or conditions for \( f'(x) \) in the interval, we can analyze those to confirm the existence of a local minimum.