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
- 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.
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.
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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.
