the graph of the derivative ( f ) of a function ( f ) is shown.\n(a) on what intervals is ( f ) increasing…

the graph of the derivative ( f ) of a function ( f ) is shown.\n(a) on what intervals is ( f ) increasing? (enter your answer using interval notation.)\non what intervals is ( f ) decreasing? (enter your answer using interval notation.)\n(b) at what values of ( x ) does ( f ) have a local maximum or local minimum? (enter your answers as a comma - separated list.)\n( x = )

the graph of the derivative ( f ) of a function ( f ) is shown.\n(a) on what intervals is ( f ) increasing? (enter your answer using interval notation.)\non what intervals is ( f ) decreasing? (enter your answer using interval notation.)\n(b) at what values of ( x ) does ( f ) have a local maximum or local minimum? (enter your answers as a comma - separated list.)\n( x = )

Answer

Explanation:

Step1: Recall the relationship between (f') and (f)

If (f'(x)>0) on an interval, then (f(x)) is increasing on that interval. If (f'(x)<0) on an interval, then (f(x)) is decreasing on that interval. Local maxima occur where (f'(x)) changes from positive to negative, and local minima occur where (f'(x)) changes from negative to positive.

Step2: Find intervals of increase and decrease

Looking at the graph of (y = f'(x)):

  • (f'(x)>0) on the intervals ([0,1)) and ((3,5)). So (f(x)) is increasing on ([0,1)\cup(3,5)).
  • (f'(x)<0) on the intervals ((1,3)) and ((5,6]). So (f(x)) is decreasing on ((1,3)\cup(5,6]).

Step3: Find local maxima and minima

  • At (x = 1), (f'(x)) changes from positive to negative, so (f(x)) has a local maximum at (x = 1).
  • At (x = 3), (f'(x)) changes from negative to positive, so (f(x)) has a local minimum at (x = 3).
  • At (x = 5), (f'(x)) changes from positive to negative, so (f(x)) has a local maximum at (x = 5).

Answer:

(a) Increasing: ([0,1)\cup(3,5)); Decreasing: ((1,3)\cup(5,6]) (b) (x = 1,3,5)