Inflection point

Definition

Let $I$ be an open interval, let $f \colon I \to \mathbb{R}$ be continuous, and let $c \in I$. The point $c$ is an inflection point of $f$ if there is $\delta > 0$ with one of the following properties. The function $f$ is convex on $(c - \delta, c]$ and concave on $[c, c + \delta)$. Or $f$ is concave on $(c - \delta, c]$ and convex on $[c, c + \delta)$.

Remarks

If $f$ is twice differentiable at an inflection point $c$, then $f''(c) = 0$. The converse fails: $f(x) = x^4$ has $f''(0) = 0$, but $0$ is not an inflection point. The function $x^3$ has an inflection point at $0$. So does $\sin x$, at each multiple of $\pi$. Some texts also ask the graph to have a tangent line at $c$.

Explanations

L1 human

At an inflection point, the graph changes from bending up to bending down, or the reverse. The graph crosses its tangent line there.