In
mathematics, a concave function is one for which the value at any convex combination of elements in the domain is greater than or equal to the convex combination of the values at the endpoints. Equivalently, a concave function is any function for which the
hypograph is convex. The class of concave functions is in a sense the opposite of the class of a
convex functions. A concave function is also
synonymously called concave downwards, concave down, convex upwards, convex cap, or upper convex.
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.[2]
If f is twice-
differentiable, then f is concave
if and only iff ′′ is
non-positive (or, informally, if the "
acceleration" is non-positive). If f ′′ is
negative then f 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:[2]
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.
^Lenhart, S.; Workman, J. T. (2007). Optimal Control Applied to Biological Models. Mathematical and Computational Biology Series. Chapman & Hall/ CRC.
ISBN978-1-58488-640-2.