the graph of ( f ) is given.\n(a) on what open intervals is ( f ) increasing?\n(b) on what open intervals is…

the graph of ( f ) is given.\n(a) on what open intervals is ( f ) increasing?\n(b) on what open intervals is ( f ) decreasing?\n(c) what are the ( x )-values of the local extrema?\n(d) on what open intervals is ( f ) concave up?\n(e) on what open intervals is ( f ) concave down?\n(f) what are the ( x )-values of the inflection points?\ncomma to separate answers as needed. )\n(b. the graph of ( f ) is not concave down on any open interval.\n(f) determine the ( x )-values corresponding to inflection points. select the correct choice and,\nif necessary, fill in the answer box to complete your choice.\n(a. the graph of ( f ) has one or more inflection points at ( x=)\n(round to the nearest integer as needed. use a comma to separate answers as needed )\n(b. the graph of ( f ) has no inflection points

Answer

Explanation:

Step1: Analyze the graph of (f')

The function (f) is increasing when (f'(x)>0), decreasing when (f'(x)<0). The concavity of (f) is related to the sign of (f''(x)). The inflection points occur where (f''(x) = 0) (i.e., where (f'(x)) has a local maximum or minimum).

Step2: Determine intervals of increase

Looking at the graph of (f'), (f'(x)>0) on ((-\infty,-14)\cup(14,\infty)). So (f) is increasing on ((-\infty,-14)\cup(14,\infty))

Step3: Determine intervals of decrease

(f'(x)<0) on ((-14,14)). So (f) is decreasing on ((-14,14))

Step4: Determine local extrema

Local maxima occur where (f') changes from positive to negative ((x = - 14)) and local minima where (f') changes from negative to positive ((x=14))

Step5: Determine concavity

The concavity of (f) is determined by the slope of (f'). (f) is concave up where (f') is increasing. Looking at the graph of (f'), (f') is increasing on ((-28,0)). So (f) is concave up on ((-28,0))

Step6: Determine concavity (continued)

(f) is concave down where (f') is decreasing. (f') is decreasing on ((0,28)). So (f) is concave down on ((0,28))

Step7: Determine inflection points

Inflection points occur where (f''(x)=0) (where (f') has a local maximum or minimum). (f') has a local maximum at (x = 0). So the inflection point is at (x = 0)

Answer:

(a) (f) is increasing on ((-\infty,-14)\cup(14,\infty))

(b) (f) is decreasing on ((-14,14))

(d) (f) is concave up on ((-28,0))

(e) (f) is concave down on ((0,28))

(f) A. The graph of (f) has one or more inflection points at (x = 0)