Increasing and Decreasing Functions

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.

Conceptual meaning

The derivative is local slope. Positive slope means outputs rise as inputs move right; negative slope means they fall. A zero derivative at one point does not by itself establish a maximum, minimum, or even a change in monotonicity.

A dependable method and decision rule

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

Fully worked example

Graphical or geometric meaning

A derivative sign chart is a compressed sketch of the graph's motion. Arrows point up on positive-sign intervals and down on negative-sign intervals. Only the sign on an entire interval, not the value at one sample point alone, supports the conclusion.

Common mistakes and why they fail

Verification and reasonableness checks

• Plot a few function values to confirm the sign chart's direction.
• Factor $f'$ and track multiplicities at its zeros.
• Verify each claimed interval lies inside the original domain.

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.

Practice

1. Where is $x^2$ decreasing?
2. Where is $x^2$ increasing?
3. Does $f'(2)=0$ alone prove an extremum?

Connections and next steps

• Critical Points
• First Derivative Test
• Curve Sketching