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.


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],

f((1-t)x+(t)y)\geq (1-t) f(x)+(t)f(y).

A function is called strictly concave if

f((1-t)x + (t)y) > (1-t) f(x) + (t)f(y)\,

for any t in (0,1) and xy.

For a function f:RR, 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 S(a)=\{x: f(x)\geq a\} are convex sets.[1]:496


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

Points where concavity changes (between concave and convex) are inflection points.

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.

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.

If f is concave and differentiable, then it is bounded above by its first-order Taylor approximation:[1]:489

f(y) \leq f(x) + f'(x)[y-x].

A continuous function on C is concave if and only if for any x and y in C

f\left( \frac{x+y}2 \right) \ge \frac{f(x) + f(y)}2.

If a function f is concave, and f(0) ≥ 0, then f is subadditive. Proof:

  • since f is concave, let y = 0, f(tx) = f(tx+(1-t)\cdot 0) \ge t f(x)+(1-t)f(0) \ge t f(x)
  • f(a) + f(b) = f \left((a+b) \frac{a}{a+b} \right) + f \left((a+b) \frac{b}{a+b} \right)
\ge \frac{a}{a+b} f(a+b) + \frac{b}{a+b} f(a+b) = f(a+b)


See also[edit]


  1. ^ a b Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0393957357. 
  2. ^ Thomas M. Cover and J. A. Thomas (1988). "Determinant inequalities via information theory". SIAM Journal on Matrix Analysis and Applications 9 (3): 384–392. doi:10.1137/0609033.