Using the Candidates Test to Determine Absolute Extrema

A relative max or min only tells you about a function's behavior near that point — it says nothing about what happens elsewhere on the interval. The absolute maximum is the single highest point on the whole domain you care about, and that point could be a critical point in the interior or it could be sitting at an endpoint where the function simply runs out of room to climb or fall. If you only check critical points, you can miss the true winner entirely.

The Candidates Test fixes this by turning the search into a finite comparison. First find every critical point inside the interval by solving f'(x) = 0 or finding where f' is undefined. Then take those x-values plus the two endpoints — this full list is your set of candidates. Plug every candidate into the original function f, not f'. Whichever output is largest is the absolute maximum value, and whichever is smallest is the absolute minimum value. No calculus judgment calls about increasing or decreasing are needed once you've built the list correctly; it's pure arithmetic comparison at the end.

This is why the test only works cleanly on a closed interval — open or infinite intervals can fail to have an absolute extremum at all, since there's no endpoint to catch the function. The Candidates Test is also the engine behind every optimization problem later in the unit: once you translate a real situation into a function on a closed interval, finding the best possible value is exactly this same process of listing candidates and comparing heights.