What is Inflection point? — Mathematics

Definition 22.13 High School Mathematics · Chapter 22 — Differentiation and Convexity

A point where the curve of $f$ crosses its tangent is an inflection point. For $f$ twice differentiable, inflection points are the points where $f''$ changes sign.

The inflection point of x x3 at the origin: the curve crosses its tangent (orange), concave on the left, convex on the right. — The inflection point of $x \mapsto x^3$ at the origin: the curve crosses its tangent (orange), concave on the left, convex on the right.

Examples

Example 22.14

$f(x) = x^3$ has $f''(x) = 6x$ : $f$ is concave on $\intoc{-\infty}{0}$ , convex on $\intco{0}{+\infty}$ , with an inflection point at the origin, where the curve crosses its tangent (the $x$ -axis).

Example 22.15 (Convexity inequalities)

The exponential is convex on $\R$ (its second derivative is itself, positive). Its tangent at $0$ is $y = 1 + x$ , so

\eu^x \geq 1 + x \quad \text{for all } x \in \R .

Such tangent line inequalities are a standard product of convexity.

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Examples

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