Medium

Increasing and Decreasing Intervals

Simple Explanation

Critical points split the x-axis into intervals. Testing the sign of f'(x) at one sample point inside each interval reveals whether f is increasing (f'>0) or decreasing (f'<0) across that whole interval.

Why Do We Need It?

Knowing exactly where a function rises and falls turns a rough sketch of a graph into an accurate one, and is the direct next step after finding critical points on the way to classifying them.

Formula

The Increasing/Decreasing Test

f'(x) > 0 ⟹ f is increasing; f'(x) < 0 ⟹ f is decreasing

The sign of the derivative directly reveals whether the original function is rising or falling at that point.

f'(x): — the derivative of f, evaluated at a specific x

When to use it: Whenever the intervals where a function increases or decreases need to be identified.

Worked Example

Find the increasing and decreasing intervals of a cubic

Using f(x) = x³ − 3x, with critical points at x=−1 and x=1, determine where f is increasing and decreasing.

Why Does This Work?

Because f'(x) = 3x²−3 is continuous, it can only change sign by passing through zero — and the zeros are exactly the critical points already found — so f' cannot change sign anywhere inside one of these intervals, making one sample point per interval enough to determine the sign for the entire interval.

Real-Life Example

A stock price's rising and falling phases

A stock's price over a year might climb, then dip during a slow quarter, then climb again.

Each of those phases is exactly an interval where the price function is increasing or decreasing, separated by the critical points where its rate of change hits zero.

Practice

f(x) = x² − 4x + 1 has a critical point at x=2. Is f increasing or decreasing on (−∞, 2)?

Medium

Common mistake

Testing the sign of f(x) itself instead of f'(x) — increasing/decreasing behavior is determined by the sign of the derivative, not the sign of the original function.

Quick Review

Critical points split the domain into test intervals.
f'(x)>0 on an interval ⟹ f increasing there; f'(x)<0 ⟹ f decreasing there.
One sample point per interval is enough, since f' cannot change sign without crossing zero.