Question

given the graph of the function f below, determine all values of x on the open interval (-9,9) where f(x) changes sign.

Explanation:

Step1: Recall the relationship between the function \( f \) and \( f^{\prime}(x) \)

The sign of \( f^{\prime}(x) \) determines where the function \( f(x) \) is increasing (\( f^{\prime}(x)>0 \)) or decreasing (\( f^{\prime}(x)<0 \)). The points where \( f^{\prime}(x) \) changes sign are the critical points of \( f(x) \) (where the graph of \( f(x) \) has a local maximum or minimum).

Step2: Analyze the graph

Looking at the graph of \( y = f(x) \), we find the \( x - \)values where the function changes from increasing to decreasing or vice - versa.

• When the function changes from increasing to decreasing (local maximum), \( f^{\prime}(x) \) changes from positive to negative.
• When the function changes from decreasing to increasing (local minimum), \( f^{\prime}(x) \) changes from negative to positive.

From the graph, we can see that \( f^{\prime}(x) \) changes sign at \( x=-6 \) and \( x = - 2 \) on the open interval \( (-9,9) \).

Answer:

The values of \( x \) where \( f^{\prime}(x) \) changes sign on the open interval \( (-9,9) \) are \( x=-6 \) and \( x=-2 \).