Alphabeta MathHow 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.
The functions , , and show that is inconclusive
Example
At , the functions , , and all have first and second derivative , but respectively have a strict minimum, a strict maximum, and no extremum.
Facts & Assumptions
Given: The three polynomial functions.
The false second-derivative claim is If , then the second derivative test still decides whether is an extremum.
The first nonzero derivative test is The first nonzero higher derivative classifies a stationary point, with power differentiation from 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.
Verification
Direct differentiation gives common first and second derivative data at the origin.
The fourth derivative is first nonzero for , with opposite signs; the third derivative is first nonzero for .
The even and odd cases of [L2] yield the three stated behaviours, explicitly realizing [L1].
Depends on
- If $f'(a)=f''(a)=0$, then the second derivative test still decides whether $a$ is an extremum
- The first nonzero higher derivative classifies a stationary point
- 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
- Monotonicity of $x \mapsto x^n$ and of $n \mapsto a^n$
- Integer powers $a^m$
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
- J. Lebl, Basic Analysis I, Taylor's theorem and related calculus (standard reference, not scraped)
- University of Minnesota MATH 5615, higher derivative test (standard reference, not scraped)