False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-01

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If $f^{'} (a) = f^{''} (a) = 0$ , then the second derivative test still decides whether $a$ is an extremum

Statement

False claim: If $f^{'} (c) = f^{''} (c) = 0$ , then the value of $f^{''} (c)$ decides whether $c$ is a local minimum, a local maximum, or neither.

Facts & Assumptions

Given: The functions $f_{+} (x) = x^{4}$ , $f_{-} (x) = - x^{4}$ , and $f_{0} (x) = x^{3}$ at $c = 0$ .

[L1]

The second derivative test is silent when $f^{''} (c) = 0$ (The second-derivative test for strict local extrema).

[L2]

The first nonzero derivative test classifies the three functions by their first nonzero derivatives (The first nonzero higher derivative classifies a stationary point).

Refutation

technique · direct

1.1

Direct differentiation gives $f_{\pm}^{'} (0) = f_{0}^{'} (0) = 0$ and $f_{\pm}^{''} (0) = f_{0}^{''} (0) = 0$ .

givenalgebra

1.2

Yet $x^{4} \geq 0$ with equality only at $0$ , so $f_{+}$ has a strict minimum; $- x^{4} \leq 0$ , so $f_{-}$ has a strict maximum; and $x^{3}$ changes sign, so $f_{0}$ has neither.

givenL2algebra

2.1

The identical second-derivative data lead to all three outcomes, refuting the claim.

step 1.1step 1.2L1∎

Depends on

Used by

The functions x⁴, -x⁴, and x³ show that f''(0)=0 is inconclusive Example

Dependency tree · two levels

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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)

If f′(a)=f′′(a)=0f′(a)=f′′(a)=0, then the second derivative test still decides whether aa is an extremum