(a)

The existence of an absolute maximum value and an absolute minimum value for  is determined by the Extreme value theorem.

Extreme value theorem :

If  is continuous on a closed interval , then  has an absolute maximum value  and an absolute minimum value  at  and  in  such that .

(b)

Closed interval method of finding absolute extreme values :

To find the absolute maximum and minimum values of a continuous function  on a closed interval , perform the following steps.

1. Find the values of  at the critical numbers of  in .

2. Find the values of  at the endpoints of the interval.

3. The largest of the values from steps  and  is the absolute maximum value and the smallest of these values is the absolute minimum value.

(a) Extreme value theorem.

(b) Closed interval method is used for finding absolute extreme values.