CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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The two-variable Hessian determinant test

Statement

Let $a$ be a critical point of a $C^2$ function of two variables, and put $A=f_{xx}(a)$ , $B=f_{xy}(a)$ , $C=f_{yy}(a)$ and $\Delta=AC-B^2$ . If $\Delta>0$ , then $A>0$ gives a strict local minimum and $A<0$ a strict local maximum. If $\Delta<0$ , there is neither. If $\Delta=0$ , this test gives no conclusion.

Facts & Assumptions

Given: $a$ is a critical point of a $C^2$ scalar field $f$ on an open subset of $\mathbb R^2$ .

[L1]

The Hessian is symmetric (The Hessian of a $C^2$ scalar field is symmetric).

[L2]

The second-derivative test classifies a critical point from definiteness or indefiniteness of its Hessian (The multivariable second-derivative test by definiteness of the Hessian).

Proof

technique · calculation

1.1

By [L1], the Hessian quadratic form is $Q(s,t)=As^2+2Bst+Ct^2$ . If $A\ne0$ , completing the square gives $Q=A(s+(B/A)t)^2+(\Delta/A)t^2$ .

L1algebra

2.1

If $\Delta>0$ , the two coefficients in step 1.1 have the sign of $A$ , so $Q$ is positive definite for $A>0$ and negative definite for $A<0$ .

step 1.1algebra

2.2

If $\Delta<0$ and $A\ne0$ , then $Q(1,0)=A$ while $Q(-B/A,1)=\Delta/A$ , which have opposite signs; hence $Q$ is indefinite. If $A=0$ , then $-B^2=\Delta<0$ , so $B\ne0$ , and $Q(1,t)=2Bt+Ct^2$ has both signs for sufficiently small positive and negative $t$ .

step 1.1algebra

3.1

Apply [L2] to steps 2.1 and 2.2. When $\Delta=0$ and $A\ne0$ , step 1.1 makes $Q=A(s+(B/A)t)^2$ , so it is semidefinite but not definite; when $A=0$ , then $B=0$ and $Q=Ct^2$ , with the same conclusion (including $Q=0$ ). Thus this is the inconclusive case of [L2].

L2step 1.1step 2.1step 2.2algebra∎

Depends on

Used by

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Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 23 results over 7 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

Analysis, Convexity, and Optimization (standard reference, not scraped)