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 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.
1. A differentiable function is concave on an interval if and only if its derivative function is monotonically decreasing on that interval, that is, a concave function has a non-increasing slope. 2. Points where concavity changes are inflection points. 3. If is twice-differentiable, then is concave if and only if is non-positive. If its second derivative is negative then it is strictly concave, but the converse is not true, as shown by. 4. If is concave and differentiable, then it is bounded above by its first-order Taylor approximation: 5. A Lebesgue measurable function on an interval is concave if and only if it is midpoint concave, that is, for any and in 6. If a function is concave, and, then is subadditive on. Proof:
Since is concave and, letting we have
For :
Functions of ''n'' variables
1. A function is concave over a convex set if and only if the function 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.
Examples
The functions and are concave on their domains, as their second derivatives and are always negative.
