Concavity
Concavity is a geometric term which describes a curve. In calculus, a graph is concave upward if the derivative, f '(x) (of the function, f(x) being graphed) is increasing upon an interval; a graph is concave downward if the derivative is decreasing. In other words, if the second derivative, f ''(x), is positive (or, if the acceleration is positive); then, the graph is concave upward; if the second derivative is negative; then, the graph is concave downward. Points where concavity changes are inflection points.
The "bottom" of a concave downward slope will have a point known as the minimal extremum; the "apex" of a concave upward slope will have a point known as the maximal extremum.
In mathematics, a function  is said to be concave on an interval  if, for all x,y in .
This is equivalent to
Additionally, is strictly concave if
Equivalently, is concave on iff the function  is convex on every subinterval of .
If is differentiable, then is concave iff  is monotone decreasing.
If is twice-differentiable, then is concave iff  is negative.
See also: convex
Referenced By
List of calculus topics | List of mathematical topics | List of mathematical topics (A-C) | List of mathematics topics
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