Maximum and Minimum Values

DEFINITION: A function has an absolute maximum (or global maximum) at if for all in where is the domain of The number is called the maximum value of on Similarly, has an absolute minimum (or global minimum) at if for all in and the number is called the minimum value of on The maximum and minimum values of are called the extreme values of

                               

DEFINITION: A function has a local maximum (or relative maximum) at if when is near [This means that for all in some open interval containing ] Similarly, has a local minimum (or relative minimum) at if when is near



THEOREM (The Extreme Value Theorem): If is continuous on a closed interval then attains an absolute maximum value and an absolute minimum value at some numbers and in


THEOREM (Fermat's Theorem): If has a local maximum or minimum at and if exists, then


DEFINITION: A critical number of a function is a number in the domain of such that either or does not exist.

REMARK: From Fermat's Theorem it follows that if has a local maximum or minimum at then is a critical number of



THE CLOSED INTERVAL METHOD
: To find the absolute maximum and minimum values of a continuous function on a closed interval :

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 Step 1 and 2 is the absolute maximum value; the smallest value of these values is the absolute minimum value.