There are two different kinds of extrema. They are absolute extrema and relative extrema.
The ablsolute extrema are the extrema of a function on an interval.
However, if there is an open interval containing c on which f(c) is a maximum,
then f(c) is called a relative maximum of f. And also If there is an open interval containing c
on which f(c) is a minimun, then f(c) is called a relative minimum.
c is a critical number when f'(c) = 0 or f is not differentible.
And f has relative extrema at x = c, when c is a critical number of f.
To find the cxtrema of a continuous function f on a closed interval [a, b],
it's good to use the following steps.
1. Find the critical numbers of f in (a, b).
2. Evaluate f at each critical number in (a, b).
3. Evaluate f at each endpoint of [a, b].
4. The least of these values is the minimum. The greatest is the maximum.
During step 2, and 3, you can also make the chart for left endpoint,
critical number, and right endpoint.
| x | Left endpoint | Critical Number | Right Endpoint |
| f(x) |
