# 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:

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

### Definition 2

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

$\ds \forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \frac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \frac {\map f {x_3} - \map f {x_2} } {x_3 - x_2}$

### Definition 3

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

$\forall x_1, x_2, x_3 \in I: x_1 < x_2< x_3: \dfrac {\map f {x_2} - \map f {x_1} } {x_2 - x_1} \ge \dfrac {\map f {x_3} - \map f {x_1} } {x_3 - x_1}$

## Geometric Interpretation

Let $f$ be a concave real function.

Then:

for every pair of points $A$ and $B$ on the graph of $f$, the line segment $AB$ lies entirely below the graph.

## 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.