Increasing and decreasing intervals

Find where f′(x) is zero or undefined, then examine its sign on the intervals separated by those values. A positive derivative gives increase and a negative derivative gives decrease. State intervals for the input x. A point with zero derivative alone does not establish a maximum or minimum; the signs on either side decide that.

The derivative can be zero at an isolated point while the function continues to increase through it. For example x³ increases through zero despite its horizontal tangent. When an exam asks where f′(x) > 0, exclude any zero-gradient input. When it asks for an interval on which the function is increasing, a closed endpoint can still belong to the interval. State your interpretation clearly.

Worked example

For f(x) = x³ − 3x, find where f′(x) is positive and negative, and classify the stationary points.

1. Differentiate: f′(x) = 3x² − 3 = 3(x − 1)(x + 1). The zeros are −1 and 1.
2. The derivative is positive for x < −1 and x > 1, and negative for −1 < x < 1.
3. At −1 the sign changes from positive to negative; at 1 it changes from negative to positive. Substitute into f.

Answer: f′ > 0 on (−∞,−1) ∪ (1,∞), f′ < 0 on (−1,1). Local maximum (−1,2); local minimum (1,−2).

Revise first: Differentiating powers, Quadratic inequalities.