TheoremStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-08-01

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The first nonzero higher derivative classifies a stationary point

Statement

Let $n \geq 2$ , and suppose there is a real $δ > 0$ such that $f$ is $n$ -times differentiable on the open interval $N_{δ} (c) = (c - δ, c + δ)$ . Suppose $f^{(j)} (c) = 0$ for $1 \leq j < n$ , while $f^{(n)} (c) \neq 0$ . 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 $N_{δ} (c)$ of The $ε$ -neighbourhood and the punctured $ε$ -neighbourhood of a point of $R$ .

[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 $\lim_{x \to c} f (x) = L \neq 0$ 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 $N$ , The canonical natural $ι (n) = n \cdot 1_{F}$ of a field, Canonical naturals are positive and strictly increasing).

Proof

technique · cases

1.1

Peano's formula gives $f (x) - f (c) = (x - c)^{n} (f^{(n)} (c) / ι (n!) + ε (x))$ , where $ε (x) \to 0$ . The parenthesized factor has the sign of $f^{(n)} (c)$ near $c$ .

L1L2L3

2.1

If $n$ is even, $(x - c)^{n} > 0$ for $x \neq c$ , 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 $n \geq 1$ is even or odd, so the cases are exhaustive.

step 2.1step 2.2cases-exhaustive∎

Depends on

Used by

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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)