Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-02
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  • 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.
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The monkey saddle x3−3xy2 has an indefinite higher-order critical point

Statement

The function f(x,y)=x3−3xy2 has a critical point with zero Hessian at the origin, but the origin is a saddle.

Facts & Assumptions

Given: f(x,y)=x3−3xy2.

[L1]

A semidefinite but not definite Hessian is inconclusive in the second-derivative test (The multivariable second-derivative test by definiteness of the Hessian).

Proof

technique · calculation
1.1

The gradient is (3x2−3y2,−6xy), and the Hessian entries are 6x,−6y,−6x; both vanish at the origin.

givenalgebra
1.2

Along the x-axis, f(t,0)=t3, which has positive and negative values arbitrarily near 0.

givenalgebra
2.1

Hence the origin is a saddle even though its Hessian is zero, illustrating the inconclusive case [L1].

step 1.1step 1.2L1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources