TheoremStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-01

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The first nonzero higher derivative classifies a stationary point

Statement

Let $n\ge2$ , and suppose there is a real $\delta>0$ such that $f$ is $n$ -times differentiable on the open interval $N_\delta(c)=(c-\delta,c+\delta)$ . Suppose $f^{(j)}(c)=0$ for $1\le j<n$ , while $f^{(n)}(c)\ne0$ . If $n$ is even, $c$ is a strict local minimum when $f^{(n)}(c)>0$ and a strict local maximum when it is negative. If $n$ is odd, $c$ is not a local extremum and $f(x)-f(c)$ changes sign at $c$ .

Facts & Assumptions

Given: The derivative hypotheses on the open neighbourhood $N_\delta(c)$ of The $\varepsilon$ -neighbourhood and the punctured $\varepsilon$ -neighbourhood of a point of $\mathbb{R}$ .

[L1]

Peano's formula is Peano's form: the normalized Taylor remainder tends to zero, with polynomial from Taylor polynomials and their remainders.

[L2]

A function tending to a nonzero real keeps its sign nearby (If $\lim_{x \to c} f(x) = L \ne 0$ then $|f| > |L|/2$ on a punctured neighbourhood of $c$ ; in particular if $L > 0$ then $f > L/2 > 0$ there); parity controls the sign of integer powers (Integer powers $a^m$ , Monotonicity of $x \mapsto x^n$ and of $n \mapsto a^n$ ).

[L3]

The factorial $n!$ is a nonzero natural, so its canonical real is positive (The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $\mathbb{N}$ , The canonical natural $\iota(n) = n \cdot 1_F$ of a field, Canonical naturals are positive and strictly increasing).

Proof

technique · cases

1.1

Peano's formula gives $f(x)-f(c)=(x-c)^n(f^{(n)}(c)/\iota(n!)+\varepsilon(x))$ , where $\varepsilon(x)\to0$ . The parenthesized factor has the sign of $f^{(n)}(c)$ near $c$ .

L1L2L3

2.1

If $n$ is even, $(x-c)^n>0$ for $x\ne c$ , so the difference has one strict sign on both sides, giving the asserted minimum or maximum.

assume-case evenstep 1.1L2

2.2

If $n$ is odd, $(x-c)^n$ has opposite signs on the two sides, so the difference changes sign and no local extremum occurs.

assume-case oddstep 1.1L2

3.1

Every natural $n\ge1$ is even or odd, so the cases are exhaustive.

step 2.1step 2.2cases-exhaustive∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 98 results over 25 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

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 lecture notes (standard reference, not scraped)
University of Minnesota MATH 5615, higher derivative test (standard reference, not scraped)