Definition:Concave Real Function/Definition 3

From ProofWiki
Jump to navigation Jump to search

Definition

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


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

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


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:

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


Also see