Definition:Concave Real Function/Definition 3

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Definition

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


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

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


Hence a geometrical interpretation: the slope of $P_1 P_2$ is greater than that of $P_1 P_3$:


ConcaveFunction3.png


Strictly Concave

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

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


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