# Calc2.46

Find the absolute minimum and maximum on of the function .

It is known that the only place an absolute minimum or maximum can occur on an interval is at one of the endpoints of the interval or at a critical point inside the interval. A critical point is a point where the function is defined and its derivative is either 0 or undefined. So start by finding the derivative.

This derivative is defined everywhere so the only possible critical numbers will be where the derivative is equal to 0.

can only be true if or but is always positive so it is never equal to . Thus the only critical point comes when . As stated before, the absolute max and min can only occur at an endpoint or at a critical number. So our only choices for here are , , or . If these are our only choices, our easiest plan is to just plug in the three values and see which gives the highest value and which gives the lowest value.

Thus, the absolute minimum comes at the point and the absolute maximum comes at the point .