(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.