Inflection point

Builds on Convex function Concave function

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 $c$ .

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Convex function definition
Concave function definition
Inflection point definition

Tag	`0Y0WS5BCJ7`
URI	`m2://0Y0WS5BCJ7`
Kind	definition
Writing	W1 stated
MSC	26A51
Defines	Inflection point
Uses	Convex function, Concave function

Inflection point

Definition

Remarks

Reading path

Further reading

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