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CorollaryStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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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 ff be continuous at an interior point cc and twice differentiable on each of (a,c)(a,c) and (c,b)(c,b). If f′′≥0f''\ge0 on one of those intervals and f′′≤0f''\le0 on the other, then cc is an inflection point of ff.

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′′≥0f''\ge0, [L1] makes ff convex; on the side where f′′≤0f''\le0, apply [L1] to −f-f to make ff 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 cc and the change of convexity/concavity meet [L2]'s definition.

step 1.1step 2.1cases-exhaustive∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 15 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources