By examining the first derivative we are able to determine for which values of x a function f(x) has a possible maximum or minimum point.

Maximum/minimum values, sometimes referred to as extreme values (extrema), fall into two categories:

Local/relative maximum/minimum - occurs at x = c if there is an open interval containing c such that f(c) ≥ f(x) [or f(c) ≤ f(x)] for all x in the interval.

Absolute/global maximum/minimum - entire domain is taken into account. These are not required to be in an open interval of the domain. (They can be endpoints.)

The Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers c and d in [a, b].

This is not the case on open intervals. For example, f(x) = x2 does not have an absolute maximum on (-2, 2).

Maximum and Minimum Values

1. Determine coordinates of endpoints.

2. Determine where f'(x) = 0.

Example: Consider the function f(x) = 3x4 - 16x3 + 18x2. Determine any local and absolute extreme values on [-1, 4] and state the coordinates where they lie.

Example: Find the absolute maximum and minimum values of the function f(x) = x3 - 3x2 + 1 -0.5 < x < 4

The First Derivative Test tells us that if c is a defined value of a continuous function f:

(a) If f' changes from positive to negative at c, then f has a local maximum at c.(b) If f' changes from negative to positive at c, then f has a local minimum at c.(c) If f' does not change sign at c, then f has no local extremum at c.

3. Apply the First Derivative Test.