Vipul at 21:18, 18 September 2010

2010-09-18T21:18:59Z

← Older revision		Revision as of 21:18, 18 September 2010
Line 10:		Line 10:

	<math>\frac{\partial^2 f}{\partial x^2} < 0 \ \forall \ x \in (a,b)</math>.		<math>\frac{\partial^2 f}{\partial x^2} < 0 \ \forall \ x \in (a,b)</math>.

			Concave functions could be of many type:

			* Concave functions that are increasing throughout, and in the limit still have positive first derivative.
			* Concave functions that are increasing throughout, and in the right limit have zero derivative.
			* Concave functions that start off increasing and end up decreasing.
			* Concave functions that start off with zero derivative and end up decreasing.
			* Concave functions that start off with negative first derivative.

	==Facts==		==Facts==

Vipul: Created page with '==Definition== ===For a function of a single variable=== Suppose $f$ is a function from an interval $(a,b)$ in the real numbers, to the real numbers. We s...'

2009-06-19T14:56:55Z

Created page with '==Definition== ===For a function of a single variable=== Suppose <math>f</math> is a function from an interval <math>(a,b)</math> in the real numbers, to the real numbers. We s...'

New page

==Definition==

===For a function of a single variable===

Suppose <math>f</math> is a function from an interval <math>(a,b)</math> in the real numbers, to the real numbers. We say that <math>f</math> is a '''concave function''' on <math>(a,b)</math> if:

<math>\frac{\partial^2 f}{\partial x^2} \le 0 \ \forall \ x \in (a,b)</math>.

We say that <math>f</math> is a '''strictly concave function''' on <math>(a,b)</math> if:

<math>\frac{\partial^2 f}{\partial x^2} < 0 \ \forall \ x \in (a,b)</math>.

==Facts==

===Negative of convex===

A function <math>f</math> is concave if and only if the function <math>-f</math> is a [[convex function]]. Similarly, <math>f</math> is a strictly concave function if and only if <math>-f</math> is a [[strictly convex function]].

===Closed under addition===

A linear combination of concave functions with ''positive'' coefficients is again concave.

===Inverse function of a concave function is concave===

If <math>f</math> is a concave function, and it is invertible, then the inverse function <math>f^{-1}</math> is also a concave function.

Concave function - Revision history

Vipul at 21:18, 18 September 2010

Vipul: Created page with '==Definition== ===For a function of a single variable=== Suppose f is a function from an interval (a,b) in the real numbers, to the real numbers. We s...'

Vipul: Created page with '==Definition== ===For a function of a single variable=== Suppose $f$ is a function from an interval $(a,b)$ in the real numbers, to the real numbers. We s...'