Summary

Relative extremum: Let c be a number in the domain of a function f. Then f(c) is a relative (or local) maximum for f if there exists an open interval (a,b) containing c such that

for all x in (a,b), and f(c) is a relative (or local) minimum for f if there exists an open interval (a,b) containing c such that

for all x in (a,b).

A function has a relative (or local) extremum (plural: extrema) at c if it has either a relative max or min there.

If c is an endpoint of the domain of f, we only consider x in the half-open interval that is in the domain.

Theorem: if a function f has a relative extremum at c, then c is a critical number or c is an endpoint of the domain.

First Derivative Test: Let c be a critical number for a function f. Suppose that f is a continuous on (a,b) and differentiable on (a,b) except possibly at c, and that c is the only critical number for f in (a,b).

1. f(c) is a relative maximum of f if the derivative f'(x) is positive in the interval (a,c) and negative in the interval (c,b).
2. f(c) is a relative minimum of f if the derivative f'(x) is negative in the interval (a,c) and positive in the interval (c,b).

Examples:

• #6, p. 292
• #8, p. 292
• #16, p. 293
• #35, p. 293
• #39, p. 294