5.4 Using the First Derivative Test to Determine Relative (Local) Extrema

Critical Points First

A critical point occurs where f'(x) = 0 or f'(x) is undefined (but f is defined there). These are the only candidates for relative extrema. Always check that the point is in the domain of f before testing it.

The First Derivative Test

At a critical point c, examine the sign of f'(x) on either side. If f' changes from positive to negative, c is a relative maximum. If f' changes from negative to positive, c is a relative minimum. If the sign doesn't change, there is no extremum there.

Reading a Sign Chart

Build a number line marking critical points, then test a value in each interval to find whether f' is positive (increasing) or negative (decreasing). The transitions across critical points reveal all relative extrema — this is essentially applying the Increasing/Decreasing Test locally.

Key terms

Critical point

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

First Derivative Test (max case)

If f' changes from positive to negative at c, f has a relative maximum at c.

First Derivative Test (min case)

If f' changes from negative to positive at c, f has a relative minimum at c.

No sign change case

If f' does not change sign at a critical point, there is no relative extremum there.

How to apply the First Derivative Test

Find critical points, test the sign of f' in intervals around each, and determine if the sign changes to classify max, min, or neither.

Try an AP-style question

f'(x) = (x − 2)(x + 1). Where does f have a relative maximum?

1. x = 1/2
2. x = 2
3. x = −1
4. x = 0