5.5 Using the Candidates Test to Determine Absolute (Global) Extrema

What the Candidates Test finds

The Candidates Test finds absolute (global) extrema of a continuous function on a closed interval [a, b]. Compare the function's values at all critical points inside the interval (where f'(x) = 0 or f'(x) is undefined) and at the endpoints a and b. The largest value is the absolute max; the smallest is the absolute min.

Why it works

By the Extreme Value Theorem, a function continuous on a closed interval must attain an absolute max and min. These extrema occur either at endpoints or at critical points — nowhere else. So checking only these finitely many candidates guarantees you find the true global extrema without needing sign analysis.

Steps to apply it

1) Confirm f is continuous on [a, b]. 2) Find f'(x); solve f'(x) = 0 and locate where f'(x) is undefined — these are critical points inside the interval. 3) Evaluate f at each critical point and at both endpoints. 4) Compare all values: the greatest is the absolute maximum, the least is the absolute minimum.

Key terms

Candidates Test

Method to find absolute extrema on [a, b]: evaluate f at all critical points in the interval and at both endpoints, then compare values.

Extreme Value Theorem

If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and minimum on that interval.

Critical point

A point in the domain of f where f'(x) = 0 or f'(x) is undefined.

Where can absolute extrema occur on [a,b]?

Only at critical points inside the interval or at the endpoints a and b.

Steps for the Candidates Test

1) Check continuity on [a,b]. 2) Find critical points. 3) Evaluate f at critical points and endpoints. 4) Compare all values to find absolute max/min.

Try an AP-style question

What are the absolute max and min of f(x) = x³ − 3x on [0, 3]?

1. Max 0, min −18
2. Max 18, min −2
3. Max 2, min −2
4. Max 18, min 0