Increasing and Decreasing Functions

A rigorous, example-driven guide to increasing and decreasing functions, including hypotheses, method choice, verification, and practice.

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Critical points and concavity changes organize the shape of a differentiable graph. A smooth curve has a local maximum, local minimum, and inflection point marked. Sign bands below the axis show intervals of increase and decrease, while curvature visibly changes near the inflection point.

The central idea

If $f$ is differentiable on an interval, $f'(x)>0$ throughout that interval implies $f$ is increasing there, while $f'(x)<0$ implies it is decreasing. Critical numbers in the domain, where $f'=0$ or $f'$ does not exist, partition the domain into intervals on which the derivative's sign can be tested.

Definitions, hypotheses, and notation

The derivative-sign implications rely on the Mean Value Theorem. If $x_1<x_2$ lie in an interval where $f'>0$, then $f(x_2)-f(x_1)=f'(c)(x_2-x_1)>0$ for some intermediate $c$, so the function is increasing. This argument shows why differentiability on the interval, not one positive sample derivative, supports the conclusion.

A derivative may equal zero at isolated points without destroying strict increase; $x^3$ is strictly increasing despite $f'(0)=0$. Sign charts use test points only after factoring or continuity of $f'$ shows the sign cannot change inside the subinterval. Repeated roots of $f'$ often preserve sign, while odd-multiplicity roots often reverse it.

A dependable method and decision rule

Determine the domain of the original function.
Compute $f'$ and find all critical numbers in that domain.
Include domain breaks as interval boundaries without calling excluded points critical numbers.
Make a sign chart by testing one point in every resulting open interval.
Translate positive intervals to increasing and negative intervals to decreasing.

Fully worked example

Common mistakes and why they fail

Sign charts describe intervals

Find where $f'$ is zero or undefined, retain values in the domain of $f$, and use them to partition that domain. Testing the sign of $f'$ on each open interval determines monotonicity. A critical point does not automatically give an extremum: if the derivative keeps the same sign, the function continues in the same direction. Endpoints still matter for absolute extrema on a closed interval even though two-sided derivatives are unavailable there. Also inspect the original formula before merging intervals. Cancellation in a simplified derivative can hide a hole or vertical asymptote inherited from $f$, and monotonicity intervals cannot cross such a domain break.