Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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The monkey saddle x3−3xy2x^3-3xy^2 has an indefinite higher-order critical point

Statement

The function f(x,y)=x3−3xy2f(x,y)=x^3-3xy^2 has a critical point with zero Hessian at the origin, but the origin is a saddle.

Facts & Assumptions

Given: f(x,y)=x3−3xy2f(x,y)=x^3-3xy^2.

[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)(3x^2-3y^2,-6xy), and the Hessian entries are 6x,−6y,−6x6x,-6y,-6x; both vanish at the origin.

givenalgebra
1.2

Along the xx-axis, f(t,0)=t3f(t,0)=t^3, which has positive and negative values arbitrarily near 00.

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

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 30 results over 10 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.

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