5.4 The First Derivative Test

The test

At a critical point c, look at how f' changes sign: + to − → local maximum, − to + → local minimum, and no sign change → neither.

Why it works

A local max sits where f stops increasing and starts decreasing (f' goes + to −); a local min is the reverse (decreasing to increasing).

Key terms

First Derivative Test

Uses the sign change of f' at a critical point.

+ to −

Local maximum.

− to +

Local minimum.

No change

Neither a max nor a min.

Meaning

Detects where f switches between increasing and decreasing.

Try an AP-style question

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

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