A. \( f^{\prime \prime}(x) \) changes sign from + to - B. \( f^{\prime \prime}(x) \) changes sign from - to + C. \( f^{\prime \prime}(x)>0 \) for all \( x \) D. \( f^{\prime \prime}(x)<0 \) for all \( x \)

To analyze the statements regarding the second derivative \( f^{\prime \prime}(x) \), we need to understand what each condition implies about the function \( f(x) \). 1. **Statement A: \( f^{\prime \prime}(x) \) changes sign from + to -** - This indicates that the second derivative is positive for some interval and then becomes negative. This typically suggests that the function \( f(x) \) is concave up in the first interval and then becomes concave down. This point where the sign changes is a point of inflection. 2. **Statement B: \( f^{\prime \prime}(x) \) changes sign from - to +** - This indicates that the second derivative is negative for some interval and then becomes positive. This suggests that the function \( f(x) \) is concave down in the first interval and then becomes concave up. This is also a point of inflection. 3. **Statement C: \( f^{\prime \prime}(x) > 0 \) for all \( x \)** - This means that the second derivative is always positive, indicating that the function \( f(x) \) is concave up for all \( x \). There are no points of inflection, and the function is always curving upwards. 4. **Statement D: \( f^{\prime \prime}(x) < 0 \) for all \( x \)** - This means that the second derivative is always negative, indicating that the function \( f(x) \) is concave down for all \( x \). Similar to statement C, there are no points of inflection, and the function is always curving downwards. In summary: - Statements A and B indicate points of inflection where the concavity of the function changes. - Statements C and D indicate consistent concavity throughout the domain of the function. If you have a specific question or need further analysis on these statements, please let me know!