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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 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) = -x4.
If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation:
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 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.
- Practical example: rays bending in computation of radiowave attenuation in the atmosphere.
- Concave polygon
- Convex function
- Jensen's inequality
- Logarithmically concave function
- Quasiconcave function
- 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.
- Varian, Hal R. (1992). Microeconomic Analysis (Third ed.). W.W. Norton and Company.