4.1 Maximum & Minimum Values
- Definitions
- Absolute (global) maximum & minimum
- A function f has an absolute (global) maximum at x = c iff f(c) >= f(x) for all x in the domain of f.
- Relative (local) maximum & minimum
- A function f has a relative (local) maximum at x = c iff f(c) >= f(x) for all x in some open interval that contains x = c.
- Important Theorems:
- Extreme Value Theorem: If f is continuous on [a, b], then f has an absolute maximum and an absolute minimum value on [a, b]
- Fermat's Theorem: If f has a relative maximum or minimum at x = c, and if f'(c) exists, then f'(c) = 0.
- Where do maxima and minima hide?
- Critical numbers (values):
- c is a critical number of f if
- f'(c) = 0 or f'(c) is undefined
- and f(c) exists
- a critical number may be an absolute or relative maximum or minimum - or something else
- Endpoints
- an endpoint may be an absolute maximum or minimum, but not a relative maximum or minimum