Math101 · Independent Ontario learning library
Math101learn.math101.caFirst Derivative Test
The first derivative test classifies critical points by tracking whether a function changes from increasing to decreasing or vice versa.
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
the function is increasing. Where
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
A stationary point without a turn
For $f(x)=x^3$, $f'(x)=3x^2$. The derivative equals zero at $0$ but is positive on both sides. The function increases through the origin, so no local maximum or minimum occurs.
This shows why solving $f'=0$ is not enough.
Points where f′ is undefined
The test also works at a critical number where the derivative does not exist. For $f(x)=|x|$, derivative signs are negative to the left and positive to the right, so the corner at $0$ is a local minimum.
Only test the point if it belongs to the original domain.
Reading a derivative graph
If a graph of $f'$ is supplied, $f$ increases where the derivative graph lies above the $x$-axis and decreases where it lies below. Crossings of the derivative axis can indicate local extrema of $f$.
Touching the axis without crossing corresponds to no sign change and usually no extremum.
Absolute versus local extrema
The first derivative test classifies local behaviour. On a closed interval, an endpoint can be the absolute maximum or minimum without being a local interior turn.
For absolute extrema, evaluate all interior critical numbers and endpoints, then compare outputs.
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.
Quick self-check
- Are all critical numbers and domain boundaries ordered?
- Is each interval labelled with the sign of $f'$?
- Do positive and negative signs translate to increasing and decreasing correctly?
- Does each critical point show $+\to-$, $-\to+$, or no change?
- Are coordinates and contextual meanings reported?
- Have endpoints been compared when absolute extrema are requested?
Related topics
Explore the idea
Tangent and accumulation explorer
Change one quantity at a time and connect what moves to First Derivative Test.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For f(x) = x³ − 3x, f′ is positive, then negative, then positive across x = −1 and x = 1. What occurs?
- At −1, f′ changes + to −, so f has a local maximum.
- At 1, f′ changes − to +, so f has a local minimum.
End of lesson
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