Topic 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema: find absolute extrema by comparing values at critical points and endpoints.

What this topic is asking

The College Board (Topic 5.5) gives the candidates test: to find the absolute (global) maximum and minimum of a continuous function on a closed interval, evaluate the function at every critical point and at both endpoints, then pick the largest and smallest values. It is the practical procedure that the Extreme Value Theorem makes possible.

The procedure

A worked candidates test

Why endpoints are candidates

On a closed interval the endpoints are genuine candidates for the absolute extremes even though they are not critical points, because the function cannot go past them. A function can be strictly increasing on $[a, b]$ with no interior critical point at all, in which case the absolute minimum is $f(a)$ and the absolute maximum is $f(b)$. Forgetting the endpoints is the single most common error on these problems. The candidates test guards against it by building the endpoints into the list every time.

Reporting the answer correctly

The exam distinguishes the value of the extremum (a $y$-value) from the location (an $x$-value). The candidates test compares $y$-values to find the extremes, but you should report both: "the absolute maximum value is $4$, attained at $x = 1$ and $x = 4$." When a question asks only for the value, give the $y$-value; when it asks where the extremum occurs, give the $x$-value. Reading the question carefully here avoids losing a point by answering with the wrong quantity. The candidates test itself is short, so the marks reward correct critical points (including any where $f'$ is undefined), correct evaluation, and a clearly stated comparison.

Why no derivative test is needed

It can feel like a shortcut to skip classifying the critical points, but the candidates test is complete on a closed interval precisely because the Extreme Value Theorem guarantees the absolute extremes exist and a theorem guarantees they occur only at critical points or endpoints. So the global maximum and minimum are certainly among the finite list of candidate values, and the largest and smallest of those values are the answers. There is no possibility of a larger value hiding elsewhere, because elsewhere is neither a critical point nor an endpoint. This is why you compare values directly rather than running a First or Second Derivative Test: classification is unnecessary when the EVT has already promised the extremes lie in your list.

Presenting the work clearly

A clean candidates-test solution lists the candidates, shows the function value at each (often in a small table), and states the comparison. Writing $f(\text{candidate})$ for every entry makes the arithmetic checkable and the conclusion obvious, and it earns the evaluation and comparison marks even if one value is computed slightly wrong. The habit of laying out all candidates, including both endpoints, in one place is what prevents the dominant error of overlooking an endpoint. Finish with an explicit sentence naming the absolute maximum and minimum values and where they occur, so the grader sees the conclusion drawn from the comparison.