Concavity of a function

In calculus, the concavity of a function describes the way in which its graph is curving (i.e. up or down).

Definition

Let $f$ be a function differentiable over an open interval $I$.

• If $f^{\prime }$ is increasing over $I$, $f$ is said to be concave up (over $I$);

• alternatively, if $f^{\prime }$ is decreasing, $f$ is said to be concave down.

A twice-differentiable function $f$ is concave up if $f^{\prime \prime }x>0$, or concave down when $f^{\prime \prime }x<0$.

If $f$ is continuous at $a$ and $f$ changes concavity at $a$, the point $(a,fa)$ is called an inflection point of $f$.

References

• openstax "derivatives and the shape of a graph" lol zotero broken sorry