Concave Function

analysis

Definition

Concave Function

A function $f:D\to \mathbb{R}$ is called concave on an interval $I\subset D$ if for all $x,y\in I$ and all $\lambda \in (0,1)$:

$$f([1-\lambda ]x+\lambda y)\ge (1-\lambda )f(x)+\lambda f(y)$$

Geometrically, this means that the line segment (secant) between any two points on the graph of the function lies below or on the graph.

Properties

• Relation to Convexity: A function $f$ is concave if and only if $-f$ is a convex function.
• Second Derivative Test: If $f$ is twice differentiable on an open interval, then $f$ is concave if and only if its second derivative is non-positive:$f^{\prime \prime }(x)\le 0$.
• Minima and Maxima: Any local maximum of a concave function is also an absolute maximum.
• Jensen’s Inequality: For a concave function $f$,$f(\mathbb{E}[X])\ge \mathbb{E}[f(X)]$.