AP Calculus AB and BC glossary

Inflection point

Also called: Point of inflection

An inflection point is a point on a graph where the concavity changes from up to down or from down to up. Finding one requires the second derivative to change sign there, which is a stronger condition than simply equalling zero.

The procedure is to find where $f''$ is zero or undefined, then build a sign chart for $f''$ and confirm an actual change of sign. The point must also be in the domain of $f$ .

Where $f$ is differentiable, an inflection point is exactly where the first derivative has a local maximum or minimum, so the graph is locally steepest or shallowest. In applied contexts it is the moment a rate stops growing and starts shrinking, or the reverse.

The mistake

Assuming $f''(c) = 0$ makes $c$ an inflection point. For $f(x) = x^4$ the second derivative is zero at the origin but stays positive on both sides, so the concavity never changes.

Appears in: Unit 5: Analytical Applications