AP Calculus AB and BC glossary

Turning point

Also called: Local turning point

A turning point is a point on a graph where the function changes from increasing to decreasing or from decreasing to increasing. The derivative must change sign there, so every turning point is a critical point, but not every critical point is a turning point.

Where f′f' is continuous, a turning point of ff shows up on the graph of f′f' as a crossing of the xx axis, not a touch. The curve f(x)=x3f(x) = x^3 has f′(x)=3x2f'(x) = 3x^2, which reaches zero at the origin but stays non-negative on both sides, so the graph flattens and keeps climbing.

A turning point does not need a derivative to exist. The graph of f(x)=∣x∣f(x) = |x| turns at the origin where f′f' is undefined, and so does the cusp of f(x)=x2/3f(x) = x^{2/3}. Direction change is what defines the point; smoothness is optional. Endpoints are the opposite case, since a closed interval endpoint can hold the absolute maximum without the graph ever turning there.

The mistake

Marking a turning point wherever the graph of f′f' has a peak or a valley. A turn in f′f' is an inflection point of ff. A turning point of ff needs f′f' to change sign there, either by crossing zero or by jumping across a point where f′f' is undefined.

Appears in: Unit 5: Analytical Applications