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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 $A = f_{x x} (a)$ , $B = f_{x y} (a)$ , $C = f_{y y} (a)$ and $Δ = A C - B^{2}$ . If $Δ > 0$ , then $A > 0$ gives a strict local minimum and $A < 0$ a strict local maximum. If $Δ < 0$ , there is neither. If $Δ = 0$ , 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 $R^{2}$ .

[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 $Q (s, t) = A s^{2} + 2 B s t + C t^{2}$ . If $A \neq 0$ , completing the square gives $Q = A (s + (B / A) t)^{2} + (Δ / A) t^{2}$ .

L1algebra

2.1

If $Δ > 0$ , 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 $Δ < 0$ and $A \neq 0$ , then $Q (1, 0) = A$ while $Q (- B / A, 1) = Δ / A$ , which have opposite signs; hence $Q$ is indefinite. If $A = 0$ , then $- B^{2} = Δ < 0$ , so $B \neq 0$ , and $Q (1, t) = 2 B t + C t^{2}$ 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 $Δ = 0$ and $A \neq 0$ , 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 $Q = C t^{2}$ , 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

The Hessian determinant of each C² holomorphic component is nonpositive; a nondegenerate critical point is a saddle Corollary

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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