Increasing & Decreasing

f' > 0 rises, f' < 0 falls — read monotonicity off the derivative.

The idea

The sign of the derivative tells us where a function rises and where it falls. Where $f'(x) > 0$ the output grows as the input grows, and $f$ is increasing; where $f'(x) < 0$ the output shrinks, and $f$ is decreasing. One sign settles a question that would otherwise mean comparing outputs two points at a time.

The points where the sign changes are where the graph turns. A continuous $f'$ cannot pass from positive to negative without passing through zero, so every turn occurs among the solutions of

$f'(x) = 0.$

The points where this holds are the critical points of $f$. A point where $f'$ does not exist also counts, since the derivative can change sign there without taking a value.

The sign of $f'$ on either side classifies each critical point. Negative then positive means $f$ falls to the point and climbs away from it — a local minimum. Positive then negative is a local maximum. If the sign does not change, the graph merely flattens for an instant and continues in the same direction: a critical point need not be a turn.

Ways to work on it

• Walkthrough. Sign of f', critical points, and classifying a minimum.
• Practice. Find the critical point of a quadratic.
• Hardest. Two critical points of a cubic and where it increases.

Not sure where to start? Take the ten-question placement test.