Find the critical numbers and absolute max and min

Question

Let f(x) = cubed root(x) (2 − x).

Find the critical numbers of the function f.

Find the absolute maximum and minimum values, if any, of f on the given interval [−1, 2].

Answer 1

(a)

Step 1 :

The function is  and the interval . 

Apply derivative on each side with respect to x .

Product rule of derivatives :

Apply the power rule: .

Step 2 :

To find the critical numbers of , equate to zero.

is in the region .

Critical point is in the given region is .

Solution :

Critcal point is .

Answer 2

(b)

Step 1 :

To find the absolute extreme of function in given region .

Substitute Left end point  in .

Substitute right end point 2 in .

Left end point is local minimum point over the interval .

Critical  point is absolute maximum point over the interval .

Solution :

Local minimum point is  .

Absolute maximum point is .