4.1 | Extrema on an Interval


The Extreme Value Theorem: If \(f\) is continuous on a closed interval \([a,b]\), then \(f\) has a minimum and a maximum on the interval.
Definition of a Critical Number Let \(f\) be defined at \(c\). If \(f'(c) = 0\) or if \(f\) is differentiable at \(c\), then \(c\) is a critical number of \(f\).
If \(f\) has a relative minimum or relative maximum at \(x = c\), then \(c\) is a critical number.
Guidelines for Finding Extrema on a Closed Interval
To find the extrema of a continuous function \(f\) on a closed interval \([a,b]\), use these steps.

  1. Find the critical numbers of \(f\) in the interval \((a,b)\).
  2. Evaluate \(f\) at each critical number in the interval \((a,b)\).
  3. Evaluate \(f\) at the endpoints of the interval \([a,b]\).
  4. The least of these values is the minimum. The greatest is the maximum.

E 4.1 Exercises