5.3 Increasing & Decreasing Intervals

Sign of f'

f is increasing where f'(x) > 0 and decreasing where f'(x) < 0. The derivative's sign controls the direction of the function.

Sign chart

To find the intervals, build a sign chart of f': mark the critical points, then test the sign of f' in each interval between them.

Key terms

Increasing

Where f'(x) > 0.

Decreasing

Where f'(x) < 0.

Method

Analyze the sign of f'(x).

Sign chart

Test f' between critical points.

Direction

The sign of f' controls increasing vs. decreasing.

Try an AP-style question

On which intervals is f(x) = x³ − 12x increasing?

1. x > 0 only
2. −2 < x < 2
3. x < −12 and x > 12
4. x < −2 and x > 2