Using the Candidates Test to Determine Absolute Global Extrema

On a closed interval, the absolute maximum and minimum have to happen somewhere, and that somewhere is either a critical point or an endpoint. Critical points are where f'(x) = 0 or f'(x) is undefined, and they catch every local peak or valley inside the interval. But a local peak only beats its immediate neighbors. It says nothing about the far edges of the interval, where the function might climb or drop even higher. That is why endpoints count as candidates too, even though no derivative test flags them.

The method is short on purpose: find every critical point in the interval, add the two endpoints, plug all of them into f(x), and compare the outputs directly. Whichever input gives the largest output is where the absolute maximum lives; the smallest output marks the absolute minimum. No reasoning about increasing or decreasing is needed at this stage, since you are just ranking actual heights on the graph. This only works cleanly on a closed, bounded interval, where you know the function actually attains its extremes.

This connects back to limits at infinity from earlier work. When an interval is not closed, like all of the real line, there may be no actual endpoint to check, so you look at what f(x) approaches as x goes to infinity or negative infinity instead. If the function keeps climbing without bound, there is no absolute maximum at all. The Candidates Test is really the closed-interval version of that same question: where does the function end up at the edges of its domain?