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.
To find the extrema of a continuous function \(f\) on a closed interval \([a,b]\), use these steps.
- Find the critical numbers of \(f\) in the interval \((a,b)\).
- Evaluate \(f\) at each critical number in the interval \((a,b)\).
- Evaluate \(f\) at the endpoints of the interval \([a,b]\).
- The least of these values is the minimum. The greatest is the maximum.