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 \(f(x)\) and \(f^{\prime}(x)\)

The sign of \(f^{\prime}(x)\) determines if \(f(x)\) is increasing (\(f^{\prime}(x)>0\)) or decreasing (\(f^{\prime}(x)<0\)). \(f^{\prime}(x)\) changes sign at the local maxima and minima of \(f(x)\) (where the function changes from increasing to decreasing or vice - versa).

Step2: Identify local maxima and minima on the graph of \(f(x)\)

• A local maximum is a point where the function changes from increasing to decreasing. A local minimum is a point where the function changes from decreasing to increasing.
• Looking at the graph of \(y = f(x)\):
• The function \(y = f(x)\) has a local maximum (changes from increasing to decreasing) at \(x=-3\) (left - hand side of the \(y\) - axis) and \(x = 3\) (right - hand side of the \(y\) - axis).
• The function \(y = f(x)\) has a local minimum (changes from decreasing to increasing) at \(x = 0\) (on the \(y\) - axis) and \(x=7\) (right - hand side of the \(y\) - axis).

Answer:

\(x=-3,0,3,7\)