A function is called strictly concave if
for any α 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.:496
4. 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.
5. 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.
6. 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) = −x4.
8. If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation::489
10. If a function f is concave, and f(0) ≥ 0, then f is subadditive. Proof:
- since f is concave, let y = 0,
- 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 not 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.
- Practical example: rays bending in computation of radiowave attenuation in the atmosphere.
- Concave polygon
- Convex function
- Jensen's inequality
- Logarithmically concave function
- Quasiconcave function
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