5.3 Determining Intervals on Which a Function is Increasing or Decreasing

Sign of f' tells increasing/decreasing

If f'(x) > 0 on an interval, then f is increasing there; if f'(x) < 0, then f is decreasing. Find critical points (where f'(x) = 0 or is undefined) to split the domain into test intervals, then check the sign of f' in each one.

Using a sign chart

Build a sign chart for f' using critical points as boundaries. Pick a test value in each interval and evaluate f' there — the sign at that point holds for the whole interval (since f' is continuous between critical points). This method connects derivative behavior to the shape of the original function's graph.

Connecting to the graph

Where f' changes from negative to positive, f has a local minimum; from positive to negative, a local maximum. Increasing/decreasing intervals describe the graph of f itself, not f' — always report answers in terms of x-values where f is increasing or decreasing.

Key terms

When is f increasing?

When f'(x) > 0 on an interval.

When is f decreasing?

When f'(x) < 0 on an interval.

Critical point

A value of x where f'(x) = 0 or f'(x) is undefined; used to split the domain into test intervals.

Sign chart method

Mark critical points on a number line, test the sign of f' in each interval to determine increasing/decreasing behavior.

f' changes from negative to positive

Indicates a local minimum of f at that critical point.

Try an AP-style question

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

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