Concave function
From Wikipedia, the free encyclopedia
In mathematics, a real-valued function f defined on an interval (or on any concave subset C of some vector space) is called concave, if for any two points x and y in its domain C and any t in [0,1], we have
A function that is concave is often synonymously called concave downward, and a function that is convex is often synonymously called concave upwards.
A function is called strictly concave if
for any t in (0,1) and x ≠ y.
A continuous function on C is concave if and only if
.
for any x and y in C. Equivalently, f(x) is concave on [a, b] if and only if the function −f(x) is convex on every subinterval of [a, b].
A differentiable function f is concave on an interval if its derivative function f ′ is monotone decreasing on that interval: a concave function has a decreasing slope. ("Decreasing" here means "non-increasing", rather than "strictly decreasing", and thus allows zero slopes.)
[edit] Properties
For a twice-differentiable function f, if the second derivative, f ′′(x), is positive (or, if the acceleration is positive), then the graph is convex; if f ′′(x) is negative, then the graph is concave. Points where concavity changes are inflection points.
If a convex (i.e., concave upward) function has a "bottom", any point at the bottom is a minimal extremum. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a maximal extremum.
If f(x) is twice-differentiable, then f(x) is concave if and only if f ′′(x) is non-positive. If its second derivative is negative then it is strictly concave, but the opposite is not true, as shown by f(x) = -x4.
A function is called quasiconcave if and only if there is an x0 such that for all x < x0, f(x) is non-decreasing while for all x > x0 it is non-increasing. x0 can also be 
, making the function non-decreasing (non-increasing) for all x. The opposite of quasiconcave is quasiconvex.
