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 f is (strictly) concave on an interval if and only if its derivative function f ′ 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 f is twice-differentiable, then f is concave if and only if f ′′ 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 f(x) = −x4.
- If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation:
- A Lebesgue measurable function on an interval C is concave if and only if it is midpoint concave, that is, for any x and y in C
- If a function f is concave, and f(0) ≥ 0, then f is subadditive on . Proof:
- Since f is concave and 1 ≥ t ≥ 0, letting y = 0 we have
- For :
- Since f is concave and 1 ≥ t ≥ 0, letting y = 0 we have
Functions of n variables
- A function f is concave over a convex set if and only if the function −f 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.