Concave function

Formally, 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

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

Also, f(x) is concave on [a, b] if and only if the function −f(x) is convex on [a, b].

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.

This definition merely states that for every z between x and y, the point (z, f(z) ) on the graph of f is above the straight line joining the points (x, f(x) ) and (y, f(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.

A differentiable function f is concave on an interval if its derivative function f ′ is monotonically 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.)

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.

If f is concave and differentiable then

${\displaystyle f(x)\leq f(y)+f'(y)[x-y]}$ ^[1]

A function is called quasiconcave if it has zero or one peaks (local maxima). More formally, 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 ${\displaystyle x}$ . Also, a function f is called quasiconvex if and only if −f is quasiconcave.

If a function f is concave, and f(0) ≥ 0, then f is subadditive. Proof:

• since f is concave, let y = 0, ${\displaystyle f(tx)=f(tx+(1-t)\cdot 0)\geq tf(x)+(1-t)f(0)\geq tf(x)}$ 
• ${\displaystyle f(a)+f(b)=f\left((a+b){\frac {a}{a+b}}\right)+f\left((a+b){\frac {b}{a+b}}\right)\geq {\frac {a}{a+b}}f(a+b)+{\frac {b}{a+b}}f(a+b)=f(a+b)}$ 

• The functions ${\displaystyle f(x)=-x^{2}}$  and ${\displaystyle f(x)={\sqrt {x}}}$  are concave, as the second derivative is always negative.
• Any linear function ${\displaystyle f(x)=ax+b}$  is both concave and convex.
• The function ${\displaystyle f(x)=\sin(x)}$  is concave on the interval ${\displaystyle [0,\pi ],\,}$ .
• The function ${\displaystyle \log |B|}$ , where ${\displaystyle |B|}$  is the determinant of matrix nonnegative-definite matrix B, is concave^[2].

• Convex function
• Quasiconcave function
• Concave polygon
• log-concave
• Jensen's inequality
• Computation of radiowave attenuation in the atmosphere

• Rao, Singiresu S. (2009). Engineering Optimization: Theory and Practice. John Wiley and Sons. p. 779. ISBN 0470183527.