CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-02

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there

Statement

Let $f$ be continuous at an interior point $c$ and twice differentiable on each of $(a, c)$ and $(c, b)$ . If $f^{''} \geq 0$ on one of those intervals and $f^{''} \leq 0$ on the other, then $c$ is an inflection point of $f$ .

Facts & Assumptions

Given: The stated continuity and one-sided twice-differentiability hypotheses.

[L1]

A twice differentiable function on an open interval is convex if and only if its second derivative is nonnegative (A twice-differentiable function on an open interval is convex if and only if its second derivative is nonnegative).

[L2]

An inflection point is a point of continuity where convexity changes to concavity or conversely (An inflection point as a point of continuity where convexity changes to concavity or conversely).

Proof

technique · cases

1.1

On the side where $f^{''} \geq 0$ , [L1] makes $f$ convex; on the side where $f^{''} \leq 0$ , apply [L1] to $- f$ to make $f$ concave.

assume-case convex_leftL1L2

2.1

If the signs occur in the opposite order, the same argument interchanges the two sides.

assume-case convex_rightstep 1.1L2algebra

3.1

In either case the assumed continuity at $c$ and the change of convexity/concavity meet [L2]'s definition.

step 1.1step 2.1cases-exhaustive∎

Depends on

Used by

x↦ x³ changes from concave to convex at zero and has an inflection point there Example

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Sources

OpenStax Calculus Volume 1, §4.5 (standard reference, not scraped)