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
    • Finding critical values on the TI-89
  • Endpoints
    • an endpoint may be an absolute maximum or minimum, but not a relative maximum or minimum