Definition:Concave Real Function

Definition
Let $f$ be a real function which is defined on a real interval $I$.

Then $f$ is concave on $I$ iff:


 * $\forall \alpha, \beta \in \R: \alpha > 0, \beta > 0, \alpha + \beta = 1: f \left({\alpha x + \beta y}\right) \ge \alpha f \left({x}\right) + \beta f \left({y}\right)$

wherever $x, y \in I$.


 * [[File:ConcaveFunction1.png]]

The geometric interpretation is that any point on the chord drawn on the graph of any concave function always lies on or below the graph.

Alternative Definition
A real function $f$ defined on a real interval $I$ is concave on $I$ iff:


 * $\displaystyle \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \frac {f \left({x_3}\right) - f \left({x_2}\right)} {x_3 - x_2}$

or:


 * $\displaystyle \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {f \left({x_2}\right) - f \left({x_1}\right)} {x_2 - x_1} \ge \frac {f \left({x_3}\right) - f \left({x_1}\right)} {x_3 - x_1}$.


 * ConcaveFunction2.png ConcaveFunction3.png

Hence a geometrical interpretation:
 * In the left hand image above, the slope of $P_1 P_2$ is greater than that of $P_2 P_3$.
 * In the right hand image above, the slope of $P_1 P_2$ is greater than that of $P_1 P_3$.

Equivalence of Definitions
These two definitions can be seen to be equivalent from Equivalence of Convex and Concave Definitions.

Note
Compare convex function. It is immediately obvious from the definition that $f$ is concave on $I$ iff $-f$ is convex on $I$.