Extrema

Definition of Maximum and Minimum (Extrema)

f(c) is the minimum of f(x) on an interval if f(c) < f(x) for all x in that interval.
f(c) is the maximum of f(x) on an interval if f(c) > f(x) for all x in that interval.

Extreme Value Theorem

If f(x) is continuous on a closed interval [a,b], then f has both a minimum and a maximum on the interval.

Relative Extrema

If there is an open interval on which f(c) is a maximum (or minimum), then f(c) is called a relative maximum (or minium) of f(x).

Absolute Extrema

THE maximum value and THE minimum value of f(x).

Critical Points

f '(c) = 0 or f '(c) is undefined.

To Find Extrema on an Interval

  1. Check where f '(x) = 0
  2. Check where f '(x) is undefined
  3. Check endpoints
  4. Evaluate f(x) at each of these points. The least of these is the absolute minimum and the greatest is the absolute maximum.
This function has an absolute minimum at x = -5 where the derivative of the function does not exist.

This function has a relative maximum at x = -2   
                                                               and a relative minimum at x = 3 where the derivative of the function is 0.

Problems