How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted — a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated — a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
changes from concave to convex at zero and has an inflection point there
Example
For , the power rule gives (For a natural the function is differentiable everywhere with derivative ; for it is the constant , with derivative ; for a natural the function is differentiable at every with derivative ; consequently every polynomial function is differentiable at every real, with the derivative computed term by term). It is negative on and positive on , while is continuous at . Therefore the sign-change criterion (A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there) makes an inflection point (An inflection point as a point of continuity where convexity changes to concavity or conversely).
Depends on
- A continuous function whose second derivative has opposite signs on the two sides of a point has an inflection point there
- An inflection point as a point of continuity where convexity changes to concavity or conversely
- For a natural $n \ge 1$ the function $x \mapsto x^{n}$ is differentiable everywhere with derivative $\iota(n)\,x^{\,n-1}$; for $n = 0$ it is the constant $1$, with derivative $0$; for a natural $n \ge 1$ the function $x \mapsto x^{-n}$ is differentiable at every $x \ne 0$ with derivative $-\iota(n)\,x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term
- Higher derivatives and the classes $C^k$ and $C^\infty$
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 55 results over 18 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
- OpenStax Calculus Volume 1, §4.5 (standard reference, not scraped)