Answer the following questions about the function whose derivative is f primef′(x)equals=x Superscript negative three fifths Baseline (x minus 3 )x− 3 5(x−3). a. What are the critical points of f? b. On what open intervals is f increasing or decreasing? c. At what points, if any, does f assume local maximum and minimum values?

(a) The critical points of f are x=0 and x=3.

(b)f is decreasing on and f is decreasing on .

(c) Therefore the local minimum of f is at x=3

Step-by-step explanation:

Given function is

(a)

To find the critical point set f'(x)=0

The critical points of f are 0,3.

(b)

The interval are and .

To find the increasing or decreasing, taking two points one point from the interval (0,3) and another point .

Assume 1 and 4.

Now

and

Since 1∈ , f'(x)<0  and 4∈ , f'(x)>0

∴f is decreasing on and f is decreasing on .

(c)

Differentiating with respect to x

Now

and

Since f''(x)>0 at x=3

Therefore the local minimum of f is at x=3