5.5 The Candidates Test

Absolute extrema on [a, b]

To find the absolute (global) max and min on a closed interval: evaluate f at all critical points in (a, b) and at both endpoints a and b.

Compare values

The largest of these f-values is the absolute maximum; the smallest is the absolute minimum. Endpoints must be included because extremes can occur at the ends.

Key terms

Candidates Test

Find absolute extrema on a closed interval.

Candidates

Critical points and endpoints.

Absolute max

The largest candidate f-value.

Absolute min

The smallest candidate f-value.

Endpoints

Must be included as candidates.

Try an AP-style question

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

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