# Concave function

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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(x)* is quasiconcave if the upper contour sets of the function are convex sets.Template:Sfn

## Properties

A function *f*(*x*) is concave over a convex set if and only if the function −*f*(*x*) 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.)

For a twice-differentiable function *f*, if the second derivative, *f ′′(x)*, is positive (or, if the acceleration is positive), then the graph is convex; if *f ′′(x)* is negative, then the graph is concave. Points where concavity changes are inflection points.

If a convex (i.e., concave upward) function has a "bottom", any point at the bottom is a minimal extremum. If a concave (i.e., concave downward) function has an "apex", any point at the apex is a maximal extremum.

If *f*(*x*) is twice-differentiable, then *f*(*x*) is concave if and only if *f* ′′(*x*) 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:

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:

## Examples

- The functions and are concave on their domains, as are 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.^{[1]} - 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

## Notes

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## References

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