Extreme Value Theorem

Continuous on a closed, bounded interval [a, b] means the max and min are always attained.

The idea

Theorem (Extreme value theorem).

If $f$ is continuous on a closed, bounded interval $[a, b]$, then $f$ attains both a maximum and a minimum on that interval: there are points $c_{\max}$ and $c_{\min}$ in $[a, b]$ with $f(c_{\min}) \leq f(x) \leq f(c_{\max}) \quad \text{for every } x \text{ in } [a, b].$

The theorem promises more than a bound. A function's values can stay below some ceiling while the function climbs toward it forever, never taking that value at any point. Attains rules this out: the maximum and minimum are values $f$ takes at actual points of the interval.

Each hypothesis is necessary. Continuity prevents the function from approaching a height and jumping away from it. A closed interval keeps the endpoint the function climbs toward inside the domain. A bounded interval prevents the function from growing without bound.

The theorem locates nothing: the extremes exist somewhere in $[a, b]$, and finding them is a separate task.

Ways to work on it

• Walkthrough. The theorem's statement plus three counterexamples — open, unbounded, discontinuous.
• Practice. Diagnose which hypothesis of the theorem fails for a given example.
• Hardest. Find the maximum and minimum of a function on a closed interval.

Not sure where to start? Take the ten-question placement test.