Concavity - (Calculus I) - Vocab, Definition, Explanations | Fiveable

Definition

Concavity describes the direction in which a curve bends. A graph is concave up if it bends upwards, and concave down if it bends downwards.

The second derivative test can determine concavity: if $f''(x) > 0$, the function is concave up; if $f''(x) < 0$, the function is concave down.
Points where the concavity changes are called inflection points.
A concave up graph resembles a U-shape, while a concave down graph resembles an upside-down U-shape.
At an inflection point, the second derivative is either zero or undefined.
Concavity affects the behavior of tangent lines and can indicate local maxima or minima.

Second Derivative: The derivative of the first derivative of a function; used to determine concavity and points of inflection.

Inflection Point: A point on a curve where the sign of the curvature (concavity) changes.

Critical Point: $x$-values where $f'(x)=0$ or $f'(x)$ does not exist; potential locations for local maxima, minima, or inflection points.