# 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.