# 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 f on an interval (or, more generally, a convex set in vector space) is said to be *concave* if, for any x and y in the interval and for any t in [0,1],

A function is called *strictly concave* if

for any t in (0,1) and *x* ≠ *y*.

For a function *f*:**R**→**R**, 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 function f is quasiconcave if the upper contour sets of the function are convex sets.^{[1]}^{:496}

## Properties

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

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.)

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

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.

If f is twice-differentiable, then f is concave if and only if f ′′ is non-positive (or, 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*) = −*x*^{4}.

If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation:^{[1]}^{:489}

A continuous function on **C** is concave if and only if for any x and y in **C**

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

- since f is concave, let
*y*= 0,

## Examples

- The functions and are concave on their domains, as their second derivatives and are always negative.
- Any affine function is both (non-strictly) concave and convex.
- The sine function is concave on the interval .
- The function , where is the determinant of a nonnegative-definite matrix
*B*, is concave.^{[2]} - Practical example: rays bending in computation of radiowave attenuation in the atmosphere.

## See also

- Concave polygon
- Convex function
- Jensen's inequality
- Logarithmically concave function
- Quasiconcave function

## References

- ↑
^{1.0}^{1.1}Varian, Hal (1992).*Microeconomic Analysis*(Third ed.). New York: Norton. ISBN 0393957357. - ↑ Thomas M. Cover and J. A. Thomas (1988). "Determinant inequalities via information theory".
*SIAM Journal on Matrix Analysis and Applications*.**9**(3): 384–392. doi:10.1137/0609033.

- Crouzeix, J.-P. (2008). "Quasi-concavity". In Durlauf, Steven N.; Blume, Lawrence E.
*The New Palgrave Dictionary of Economics*(Second ed.). Palgrave Macmillan. doi:10.1057/9780230226203.1375. - Rao, Singiresu S. (2009).
*Engineering Optimization: Theory and Practice*. John Wiley and Sons. p. 779. ISBN 0-470-18352-7.