suppose that the function graphed below is ( f(x) ), the derivative of ( f(x) ). find the locations of all…

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.\nselect the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. 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.)\nb. 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.)\nc. 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.)\nd. the function ( f(x) ) has no relative extrema.\ne. there is not enough information given.

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.\nselect the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.\na. 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.)\nb. 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.)\nc. 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.)\nd. the function ( f(x) ) has no relative extrema.\ne. there is not enough information given.

Answer

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)