Concave function

In mathematics, a real-valued function f defined on an interval (or on any convex set 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

${\displaystyle f(tx+(1-t)y)\geq tf(x)+(1-t)f(y).}$

A function that is concave is often synonymously called concave down, and a function that is convex is often synonymously called concave up.

A function is called strictly concave if

${\displaystyle f(tx+(1-t)y)>tf(x)+(1-t)f(y)\,}$

for any t in (0,1) and x ≠ y.

A continuous function on C is concave if and only if

${\displaystyle f\left({\frac {x+y}{2}}\right)\geq {\frac {f(x)+f(y)}{2}}}$ .

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.)

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 ${\displaystyle x_{0}}$  such that for all ${\displaystyle x<x_{0}}$ , ${\displaystyle f(x)}$  is non-decreasing while for all ${\displaystyle x>x_{0}}$  it is non-increasing. ${\displaystyle x_{0}}$  can also be ${\displaystyle \pm \infty }$ , making the function non-decreasing (non-increasing) for all ${\displaystyle x}$ . The opposite of quasiconcave is quasiconvex.

Here's the proof for the fact that the derivative of a concave function is decreasing: Let x < y < z. Then u(y) = u�z − y z − x x + y − x z − x z�� z − y z − x u(x) + y − x z − x u(z) or equivalently (z − x)u(y) � (z − y)u(x) + (y − x)u(z) . This implies immediately u(y) − u(x) y − x � u(z) − u(x) z − x � u(z) − u(y) z − y . which is what we wanted to prove

See also

• Convex function
• Quasiconcave function
• Concave polygon
• log-concave