Convexity, concavity and inflection points

While the first derivative describes increase, the second derivative describes the curvature of the graph: whether the function “holds water” ($\cup$) or “pours it out” ($\cap$).

Definition — Convexity, concavity and inflection

A twice-differentiable function $f$ is:

• convex (“belly up”, $\cup$) on an interval if $f''(x)\ge 0$ for every $x$ in the interval;
• concave (“belly down”, $\cap$) if $f''(x)\le 0$.

An inflection point is a point where the function changes concavity: $f''$ goes from positive to negative (or vice versa).

At the inflection point the concavity changes: to the left the curve is convex ($f''>0$), to the right concave ($f''<0$).

Links

Topics: Function study
Concepts: Concavity and convexity · Second derivative · Inflection
Skills: Studying a function · Sketching a graph