Increasing and Decreasing Intervals — Cheat sheet

CHEAT SHEET · FREE

Key method

Strict increase means $x_1<x_2\Rightarrow f(x_1)<f(x_2)$ for every pair in the interval. Strict decrease reverses the output inequality. Later, derivative signs provide an efficient test.

Example

For $f(x)=(x-1)^2$ , the graph decreases toward its vertex on $(-\infty,1]$ and increases away from it on $[1,\infty)$ . The values on both intervals are nonnegative, illustrating why positive output does not mean increasing.

Avoid this mistake

A flat tangent at one point does not automatically mean a change in monotonicity. For example, $x^3$ increases through zero.