Concavity

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.

5 Must Know Facts For Your Next Test

1. 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.
2. Points where the concavity changes are called inflection points.
3. A concave up graph resembles a U-shape, while a concave down graph resembles an upside-down U-shape.
4. At an inflection point, the second derivative is either zero or undefined.
5. Concavity affects the behavior of tangent lines and can indicate local maxima or minima.

Review Questions

• How do you determine whether a function is concave up or concave down?
• What are inflection points and how are they related to concavity?
• Describe the shape of a graph that is concave up versus one that is concave down.

Related terms

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.