- If
*f(c) ≥ f(x)*for all*x*in the domain of*f*, then*f(c)*is called the**absolute maximum**of*f*. - If
*f(c) ≤ f(x)*for all*x*in the domain of*f*, then*f(c)*is called the**absolute minimum**of*f*. - An absolute maximum or absolute minimum is called an
**absolute extremum**.

A function *f* that is continuous on a closed interval [*a, b*] has both an absolute maximum and an absolute minimum on that interval.

Absolute extrema (if they exist) must occur at critical numbers or at endpoints.

- Check to make certain that
*f*is continuous over [*a, b*]. - Find the critical numbers in the interval (
*a, b*). - Evaluate
*f*at the endpoints*a*and*b*and at the critical numbers found in step 2. - The absolute maximum of
*f*on [*a, b*] is the largest value found in step 3. - The absolute minimum of
*f*on [*a, b*] is the smallest value found in step 3.

Let *c* be a critical number of *f(x)* such that *f'(c) = 0*. If the second derivative *f''(c) > 0*, then *f(c)* is a local minimum. If *f''(c) < 0*, then *f(c)* is a local maximum.

f'(c) | f''(c) | Graph of f is: | f(c) |
---|---|---|---|

0 | + | Concave upward | Local minimum |

0 | - | Concave downward | Local maximum |

0 | 0 | ? | Test does not apply |

Let *f* be continuous on an open interval *I* with only one critical number *c* in *I*.

- If
*f'(c) = 0*and*f''(c) > 0*, then*f(c)*is the absolute minimum of*f*on*I*. - If
*f'(c) = 0*and*f''(c) < 0*, then*f(c)*is the absolute maximum of*f*on*I*.