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.

Contents

DefinitionEdit

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  ,[1]

 

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.[2]:496

PropertiesEdit

Functions of a single variableEdit

1. 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.

2. Points where concavity changes (between concave and convex) are inflection points.

3. 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 opposite is not true, as shown by f(x) = −x4.

4. If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation:[2]:489

 

5. 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

 

6. 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  :
 

Functions of n variablesEdit

1. A function f is concave over a convex set if and only if the function −f is a convex function over the set.

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.

3. Near a 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.

4. Any local maximum of a concave function is also a global maximum. A strictly concave function will have at most one global maximum.

ExamplesEdit

  • 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.[3]

ApplicationsEdit

See alsoEdit

ReferencesEdit

  1. ^ Lenhart, S.; Workman, J. T. (2007). Optimal Control Applied to Biological Models. Mathematical and Computational Biology Series. Chapman & Hall/ CRC. ISBN 978-1-58488-640-2.
  2. ^ a b Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0-393-95735-7.
  3. ^ Cover, Thomas M.; Thomas, J. A. (1988). "Determinant inequalities via information theory". SIAM Journal on Matrix Analysis and Applications. 9 (3): 384–392. doi:10.1137/0609033.
  4. ^ Pemberton, Malcolm; Rau, Nicholas (2015). Mathematics for Economists: An Introductory Textbook. Oxford University Press. pp. 363–364. ISBN 978-1-78499-148-7.

Further ReferencesEdit