Question

suppose that the function graphed below is ( f(x) ), the derivative of ( f(x) ). find the locations of all relative extrema of ( f(x) ), and tell whether each extremum is a relative maximum or minimum.
select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
a. the function ( f(x) ) has a relative maximum at ( x=square ) and has no relative minimum. (round to the nearest integer as needed. use a comma to separate answers as needed.)
b. the function ( f(x) ) has a relative minimum at ( x=square ) and has a relative maximum at ( x=square ). (round to the nearest integer as needed. use a comma to separate answers as needed.)
c. the function ( f(x) ) has a relative minimum at ( x=square ) and has no relative maximum. (round to the nearest integer as needed. use a comma to separate answers as needed.)
d. the function ( f(x) ) has no relative extrema.
e. there is not enough information given.

Explanation:

Step1: Recall the first - derivative test

The first - derivative test states that if the derivative of a function \(y = f(x)\) changes sign from positive to negative at a critical point \(x = c\), then \(f(x)\) has a relative maximum at \(x = c\). If the derivative changes sign from negative to positive at a critical point \(x = c\), then \(f(x)\) has a relative minimum at \(x = c\).

Step2: Analyze the graph of \(y = f^{\prime}(x)\)

Looking at the graph of \(y=f^{\prime}(x)\), we find the \(x\) - values where \(f^{\prime}(x)=0\) (the \(x\) - intercepts of \(y = f^{\prime}(x)\)).
We observe that \(f^{\prime}(x)\) changes sign from positive to negative at \(x=-12\) (since to the left of \(x = - 12\), \(f^{\prime}(x)>0\) and to the right of \(x=-12\), \(f^{\prime}(x)<0\)). So, by the first - derivative test, \(f(x)\) has a relative maximum at \(x=-12\).
We also observe that \(f^{\prime}(x)\) changes sign from negative to positive at \(x = 16\) (since to the left of \(x = 16\), \(f^{\prime}(x)<0\) and to the right of \(x = 16\), \(f^{\prime}(x)>0\)). So, by the first - derivative test, \(f(x)\) has a relative minimum at \(x = 16\).

Answer:

B. The function \(f(x)\) has a relative minimum at \(x = 16\) and has a relative maximum at \(x=-12\)