Increasing and decreasing functions

The essentials

A function f is strictly increasing on an interval if larger inputs always give larger outputs there, and strictly decreasing if larger inputs always give smaller outputs. The first derivative gives a quick test. If f′(x) > 0 at every point of an open interval, f is strictly increasing on it; if f′(x) < 0 throughout, f is strictly decreasing; if f′(x) = 0 throughout, f is constant.

To find the intervals, solve f′(x) = 0 to get the critical points. These split the real line into intervals; pick any test value in each interval and check the sign of f′ there.

Key points

• Linear f(x) = mx + c is strictly increasing if m > 0 and strictly decreasing if m < 0.
• f(x) = x² is strictly decreasing on (−∞, 0) and strictly increasing on (0, ∞).
• f(x) = x³ is strictly increasing on R even though f′(0) = 0; an isolated zero of f′ does not spoil monotonicity.
• eˣ is strictly increasing on R; log x is strictly increasing on (0, ∞).
• Factorise f′(x) to see its sign quickly, for example 6(x − 1)(x − 2).

Exam tips

• Draw a sign chart for f′(x) on a number line with the critical points marked.
• For "find a so that f is increasing on R", require f′(x) ≥ 0 for all x, for example by making a quadratic's discriminant non-positive or its constant term non-negative.

Common mistakes

• Reporting critical points as the answer instead of the intervals between them.
• Assuming a function increasing on two separate intervals is increasing on their union without checking.