AP Calculus AB and BC

Critical Point vs Extremum

Q: Is every critical point a maximum or a minimum?

No. For f(x) = x^3 the origin is critical and is neither, because f' does not change sign there.

Critical points are candidates; extrema are verdicts. Every interior extremum sits at a critical point, but x cubed has one at the origin and no extremum. A first derivative sign change promotes a candidate to a local extremum; comparing values at the candidates and the endpoints promotes one to an absolute extremum.

Critical point

Use when: You are gathering candidates, by finding where the derivative is zero and where it fails to exist while the function is still defined.

Extremum

Use when: You are naming an actual maximum or minimum, which takes a sign test, a second derivative check, or a comparison of values.

Side by side

	Critical point	Extremum
Definition	$f'(c) = 0$ or $f'(c)$ does not exist, with $c$ an interior point of the domain	$f(c)$ is the largest or smallest value on the interval considered
Role in the problem	Candidate	Conclusion
How you get it	Solve $f'(x) = 0$ and locate where $f'$ is undefined	Test the sign of $f'$ across the candidate, or compare values
Can sit at an endpoint	No, a critical point is interior by convention	Yes, an absolute extremum can occur at an endpoint
Counterexample	$f(x) = x^3$ at $x = 0$ : critical, not an extremum	None; an interior extremum is always at a critical point

The one-way implication behind this page is Fermat's theorem: if $f$ has a local extremum at an interior point $c$ and $f'(c)$ exists, then $f'(c) = 0$ . That guarantees the critical points contain every interior extremum, so the search is complete. It says nothing in the other direction, so a critical point is only a place worth checking.

$f(x) = x^3$ is the standing counterexample. Its derivative $f'(x) = 3x^2$ is zero at the origin, so the origin is a critical point, but $f'$ is positive on both sides of it. The function is increasing straight through, so the slope flattens without the graph ever turning around.

$f'$ goes from negative to positive at $c$ : local minimum
$f'$ goes from positive to negative at $c$ : local maximum
$f'$ keeps its sign through $c$ : no extremum, just a flat spot or a corner

Do not miss the undefined ones

A point where $f'(c)$ fails to exist is still critical, and it can still be an extremum. For $f(x) = \lvert x \rvert$ the origin has no derivative and is a genuine minimum, so a search that only solves $f'(x) = 0$ walks right past it.

Frequently asked questions

Is every critical point a maximum or a minimum?

No. For $f(x) = x^3$ the origin is critical and is neither, because $f'$ does not change sign there.

Can an extremum occur where the derivative does not exist?

Yes, and that point is still a critical point. The absolute value function has a minimum at the origin, where its derivative is undefined.

Does a critical point have to be in the domain of the function?

Yes, and it has to be an interior point of that domain. For $f(x) = \frac{1}{x}$ the derivative is undefined at $x = 0$ , but so is the function, so $x = 0$ is not a critical point.

In the CED: Unit 5: Analytical Applications