First Derivative Test

The first derivative test classifies critical points by tracking whether a function changes from increasing to decreasing or vice versa.

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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 sign of $f'$ tells whether the original graph rises or falls; a sign change tells whether it turns.

Increasing and decreasing

On an interval where

$$ f'(x)>0, $$

the function is increasing. Where

$$ f'(x)<0, $$

it is decreasing.

The derivative's sign, rather than its size alone, determines the direction of motion along the graph.

The classification rules

At a critical number $c$:

if $f'$ changes from positive to negative, $f$ has a local maximum;
if $f'$ changes from negative to positive, $f$ has a local minimum;
if $f'$ has the same sign on both sides, there is no local extremum.

The test explains the turn by describing behaviour before and after the point.

Building a sign chart

Find critical numbers and domain breaks.
Place them in order on a number line.
Choose one test input from each interval.
Determine the sign of $f'$ there.
Translate signs into increasing/decreasing arrows.
Classify sign changes.

A factored derivative often makes signs easier to read.

Worked example

Context interpretation

A positive derivative may mean increasing position, cost, population, or temperature. A local maximum may represent peak height or revenue, but only if it lies inside the model's realistic domain.

State the input and output meanings rather than reporting “max” without context.

Common mistakes

Testing signs of $f$ instead of $f'$. The derivative controls increasing/decreasing behaviour.

Classifying from $f'(c)=0$ alone. Check both sides.

Ignoring domain breaks. They split intervals even when not critical points.

Calling a positive-to-negative change a minimum. It rises then falls, so it is a maximum.

Using local tests to claim an absolute result without endpoint comparison. The tasks differ.