# Definition:Concave Real Function

## Definition

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

### Definition 1

$f$ is concave on $I$ if and only if:

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

### Definition 2

$f$ is concave on $I$ if and only if:

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

### Definition 3

$f$ is concave on $I$ if and only if:

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

## Also known as

A concave function can also be referred to as:

• a concave down function
• a convex up function

## Also see

• Results about concave real functions can be found here.