Extreme Value Theorem, Extrema, and Critical Points

Picture a continuous curve trapped between two closed ends, like a fence on both sides. The curve can wiggle however it wants, but it cannot escape to infinity and it cannot jump over a height without passing through it. That trapping is why a highest point and a lowest point must exist somewhere on the interval. This is the Extreme Value Theorem: continuous function, closed interval, guaranteed absolute max and absolute min.

Knowing the extremes exist is only half the job — you still have to find them. The theorem narrows the search: every extreme value happens either at an endpoint of the interval or at a critical point inside it, where f'(x) = 0 or f'(x) is undefined. So the method is mechanical. List the critical points, add the two endpoints, evaluate f at each one, and compare. The largest output is the absolute max, the smallest is the absolute min. Nothing else needs checking, because continuity rules out any surprise extreme hiding between those points.

This connects directly to what comes next: not every critical point is actually a max or min, which is why the next step is testing each one with the first or second derivative. It also explains why the closed interval matters so much — remove one endpoint and the guarantee can break, because the curve might creep toward a value without ever reaching it. The Extreme Value Theorem is the reason the whole optimization process in this unit is allowed to work at all.