AP Calculus AB and BC

Critical Point vs Inflection Point

Q: Can a point be both critical and an inflection point?

Yes. For f(x) = x^3 the origin has zero slope and a concavity change, so it is both, and it is not an extremum.

A critical point is where the first derivative is zero or undefined, and it is where extrema can occur. An inflection point is where the second derivative changes sign, and it is where concavity flips. They answer different questions and often sit at different places.

Critical point

Use when: You are hunting maxima and minima, or describing where the function turns around.

Inflection point

Use when: You are describing where the graph changes its bend, or where a rate stops growing and starts shrinking.

Side by side

	Critical point	Inflection point
Derivative involved	$f'$	$f''$
Condition	$f'$ is zero or undefined	$f''$ changes sign
Tells you about	Extrema	Concavity
Zero is enough	Yes, for candidacy	No, the sign must change

The two are independent. A function can have a critical point that is not an inflection point, as $x^2$ does at the origin, and an inflection point that is not critical, as $x^3 - 3x$ does at the origin where the slope is $-3$ .

Both come with the same trap in different forms. For a critical point, a zero derivative makes the point a candidate but not automatically an extremum. For an inflection point, a zero second derivative is not enough either, because the sign must actually change.

The classic counterexample

For $f(x) = x^4$ the second derivative is zero at the origin but stays positive on both sides, so the concavity never changes and there is no inflection point there.

Frequently asked questions

Can a point be both critical and an inflection point?

Yes. For $f(x) = x^3$ the origin has zero slope and a concavity change, so it is both, and it is not an extremum.

In the CED: Unit 5: Analytical Applications