Question

suppose that the function graphed below is ( f^{prime}(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 minimum at ( x=square ) (round to the nearest integer as needed. use a comma to separate answers as needed.) and has a relative maximum at
( x=square ). (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 ) (round to the nearest integer as needed. use a comma to separate answers as needed.) and has no relative maximum.
c. the function ( f(x) ) has a relative maximum at ( x=square ) (round to the nearest integer as needed. use a comma to separate answers as needed.) and has no relative minimum.
d. the function ( f(x) ) has no relative extrema.
e. there is not enough information given.

Explanation:

Step1: Identify relative extrema

Relative extrema occur where the function changes from increasing to decreasing (relative maximum) or vice - versa (relative minimum).

Step2: Analyze the graph

Looking at the graph, we can see that there are points where the function changes its direction.

Step3: Determine the \(x\) - values

By observing the \(x\) - axis of the graph (assuming the standard coordinate system where \(x\) is the horizontal axis), we can estimate the \(x\) - values of the relative extrema.

Step4: Classify as maximum or minimum

At the left - hand turning point (where the function changes from increasing to decreasing), it is a relative maximum. At the right - hand turning point (where the function changes from decreasing to increasing), it is a relative minimum.

Answer:

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