Concave function

Concave function
In mathematics, a concave function is the negative of a convex function. A concave function is also synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
Definition
A real-valued function on an interval (or, more generally, a convex set in vector space) is said to be concave if, for any and in the interval and for any ,
A function is called strictly concave if
for any and .
For a function , this second definition merely states that for every strictly between and , the point on the graph of is above the straight line joining the points and .
A function is quasiconcave if the upper contour sets of the function are convex sets.
Properties
Functions of a single variable
A differentiable function is (strictly) concave on an interval if and only if its derivative function is (strictly) monotonically decreasing on that interval, that is, a concave function has a non-increasing (decreasing) slope.
Points where concavity changes (between concave and convex) are inflection points.
If is twice-differentiable, then is concave if and only if is non-positive (or, informally, if the "acceleration" is non-positive). If its second derivative is negative then it is strictly concave, but the converse is not true, as shown by .
If is concave and differentiable, then it is bounded above by its first-order Taylor approximation:
A Lebesgue measurable function on an interval is concave if and only if it is midpoint concave, that is, for any and in
If a function is concave, and , then is subadditive on . Proof:
Since is concave and , letting we have
For :
Functions of n variables
A function is concave over a convex set if and only if the function is a convex function over the set.
The sum of two concave functions is itself concave and so is the pointwise minimum of two concave functions, i.e. the set of concave functions on a given domain form a semifield.
Near a strict local maximum in the interior of the domain of a function, the function must be concave; as a partial converse, if the derivative of a strictly concave function is zero at some point, then that point is a local maximum.
Any local maximum of a concave function is also a global maximum. A strictly concave function will have at most one global maximum.
Examples
The functions and are concave on their domains, as their second derivatives and are always negative.
The logarithm function is concave on its domain , as its derivative is a strictly decreasing function.
Any affine function is both concave and convex, but neither strictly-concave nor strictly-convex.
The sine function is concave on the interval .
The function , where is the determinant of a nonnegative-definite matrix B, is concave.
Applications
Rays bending in the computation of radiowave attenuation in the atmosphere involve concave functions.
In expected utility theory for choice under uncertainty, cardinal utility functions of risk averse decision makers are concave.
In microeconomic theory, production functions are usually assumed to be concave over some or all of their domains, resulting in diminishing returns to input factors.
See also
Concave polygon
Jensen's inequality
Logarithmically concave function
Quasiconcave function
Concavification
References
Further References
Convex analysis
Types of functions