The twice-differentiable function \( f \) is shown below on the domain \( (-9,9) \). The function \( f \) has points of inflection at \( x=-7.5, x=-2 ; x=1.4, x=7.4 \), shown with small green circies on the graph. Determine what could be said about the values of \( f(-2), f^{\prime}(-2) \), and \( f^{\prime \prime}(-2) \).

To analyze the behavior of the function \( f \) at the point \( x = -2 \), where there is a point of inflection, we can use the following properties of twice-differentiable functions: 1. **Point of Inflection**: A point of inflection occurs where the second derivative \( f''(x) \) changes sign. This means that at \( x = -2 \), the second derivative \( f''(-2) \) is equal to zero, but it is not sufficient to conclude that \( f''(-2) \) is positive or negative. 2. **First Derivative**: The first derivative \( f'(-2) \) can be either positive, negative, or zero. The presence of a point of inflection does not provide specific information about the value of the first derivative at that point. It could indicate a local maximum, minimum, or neither. 3. **Function Value**: The value of the function \( f(-2) \) is simply the output of the function at that point. The point of inflection does not impose any specific condition on the value of the function itself. Based on this analysis, we can summarize the information about \( f(-2) \), \( f'(-2) \), and \( f''(-2) \) as follows: - \( f(-2) \): The value of the function at \( x = -2 \) is not determined by the point of inflection. - \( f'(-2) \): The value of the first derivative at \( x = -2 \) can be any real number (positive, negative, or zero). - \( f''(-2) = 0 \): The second derivative at \( x = -2 \) is zero, indicating a change in concavity. In conclusion: - \( f(-2) \) is undetermined. - \( f'(-2) \) can be any real number. - \( f''(-2) = 0 \).