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 pointss.

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.