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) ).
2. 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.)
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.
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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