Question

the graph of ( f ) is given. assume that ( f ) is continuous and determine the ( x )-values corresponding to local minima and local maxima. determine the ( x )-value(s) corresponding to local maxima. select the correct choice below and, if necessary, fill in the answer box to complete your choice. a. the local maximum(s) is/are located at ( x = ) (round to the nearest integer as needed. use a comma to separate answers as needed ) b. there is no local maximum.

Explanation:

Step1: Recall the first - derivative test

If \(f^{\prime}(x)\) changes sign from positive to negative at a critical point \(x = c\), then \(f(x)\) has a local maximum at \(x = c\). Critical points occur where \(f^{\prime}(x)=0\) or \(f^{\prime}(x)\) is undefined.

Step2: Analyze the sign change of \(f^{\prime}(x)\)

Looking at the graph of \(y = f^{\prime}(x)\), we find the \(x\) - values where \(f^{\prime}(x)\) crosses the \(x\) - axis. At \(x=-6\), the function \(y = f^{\prime}(x)\) changes from positive (the function \(y = f^{\prime}(x)\) is above the \(x\) - axis) to negative (the function \(y = f^{\prime}(x)\) is below the \(x\) - axis).

Answer:

A. The local maximum(s) is/are located at \(x=-6\)