Second Derivative Test for Multivariable Functions

Core Definition

Let $z = f(x, y)$ have continuous second-order partial derivatives on an open disk centered at a critical point $(a, b)$ where $\nabla f(a, b) = \mathbf{0}$ . The Hessian determinant (or discriminant) $D$ is defined by:

D(x, y) = \det \begin{bmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{bmatrix} = f_{xx}f_{yy} - (f_{xy})^2

Evaluating $D(a, b)$ and $f_{xx}(a, b)$ yields four distinct geometric classifications:

Local Minimum: If $D(a, b) > 0$ and $f_{xx}(a, b) > 0$ (or $f_{yy}(a, b) > 0$ ).
Local Maximum: If $D(a, b) > 0$ and $f_{xx}(a, b) < 0$ (or $f_{yy}(a, b) < 0$ ).
Saddle Point: If $D(a, b) < 0$ .
Inconclusive Test: If $D(a, b) = 0$ (the critical point may be a local minimum, local maximum, saddle point, or flat trough; higher-order analysis or geometric inspection is required).

Concept Explanation

Geometric and Curvature Foundations

The Second Derivative Test analyzes the quadratic curvature of the Taylor expansion of $f(x, y)$ near the critical point $(a, b)$ :

f(a + h, b + k) \approx f(a, b) + \frac{1}{2}\left[ f_{xx}h^2 + 2f_{xy}hk + f_{yy}k^2 \right]

The bracketed term is a quadratic form $Q(h, k) = A h^2 + 2B hk + C k^2$ , with matrix:

H = \begin{bmatrix} f_{xx} & f_{xy} \\ f_{xy} & f_{yy} \end{bmatrix}

(where

f_{xy} = f_{yx}

by Clairaut's Theorem).

Completing the square in the quadratic form reveals why the discriminant $D = AC - B^2$ dictates surface geometry:

$D > 0$ (Definite Curvature — Bowl Shape):
- Since $f_{xx}f_{yy} > (f_{xy})^2 \ge 0$ , both pure second derivatives $f_{xx}$ and $f_{yy}$ must share the same non-zero sign.
- If $f_{xx} > 0$ , the surface is concave upward in all vertical cross-sections, forming a bowl. The point is a local minimum.
- If $f_{xx} < 0$ , the surface is concave downward in all vertical cross-sections, forming an inverted dome. The point is a local maximum.
$D < 0$ (Indefinite Curvature — Saddle Shape):
- The quadratic form takes on both positive and negative values depending on the direction of travel $\langle h, k \rangle$ .
- Along one trajectory, the cross-section curves upward ( $f$ increases); along a transverse trajectory, the cross-section curves downward ( $f$ decreases).
- The point is a saddle point (resembling a mountain pass or a horse saddle).

    Local Minimum (D > 0, f_xx > 0)          Saddle Point (D < 0)
              \       /                           .-' \   / '-.
               \  *  /                           /     \ /     \
                '---'                           (   *   x   *   )
               Bowl Shape                        \     / \     /
                                                  '-. /   \ .-'
                                                   Mountain Pass

Why $f_{xx}$ Determines the Sign When $D > 0$

When $D = f_{xx}f_{yy} - (f_{xy})^2 > 0$ , we have $f_{xx}f_{yy} > (f_{xy})^2 \ge 0$ . This forces $f_{xx}$ and $f_{yy}$ to have the exact same algebraic sign. Hence, checking $f_{xx} > 0$ is mathematically equivalent to checking $f_{yy} > 0$ .

Step-by-Step Procedure

Find all critical points $(a, b)$ satisfying $f_x = 0$ and $f_y = 0$ (using Critical Points in Multivariable Functions).
Compute second-order partial derivatives: $f_{xx}$ , $f_{yy}$ , and $f_{xy}$ .
Formulate the general discriminant equation: $D(x, y) = f_{xx}(x, y)f_{yy}(x, y) - [f_{xy}(x, y)]^2$
For each critical point $(a, b)$ :
- Calculate the numerical value of $D(a, b)$ .
- If $D(a, b) > 0$ , evaluate the sign of $f_{xx}(a, b)$ .
- Classify according to the discriminant rules.

Engineering Applications

Structural Mechanics: In finite element analysis (FEA), elastic potential energy fields must possess positive-definite Hessians ( $D > 0, f_{xx} > 0$ ) at loaded equilibrium points to ensure dynamic stability; negative discriminants indicate buckling or snapping modes.
Aerospace Engineering: In aircraft wing design, the lift-to-drag aerodynamic efficiency surface $E(\alpha, \beta)$ is optimized. Engineers seek a true local maximum ( $D > 0, E_{\alpha\alpha} < 0$ ) to guarantee robust cruise performance against perturbations in angle of attack $\alpha$ or flap deflection $\beta$ .
Electrical & Audio Engineering: Audio dynamic amplifiers use automatic gain control loops where stable operating quiescent points require positive definite matrices, preventing harmonic distortion or self-sustaining oscillations.

Example Problems

Example 1: Multi-Point Classification (Local Minimum and Saddle Point)

Find and classify all critical points of:

f(x, y) = 2x^3 - 6x + 2y^2 - 4y

Solution

First-order partial derivatives:
$f_x = 6x^2 - 6, \quad f_y = 4y - 4$
Critical points:
$6x^2 - 6 = 0 \implies x^2 = 1 \implies x = \pm 1$ $4y - 4 = 0 \implies y = 1$ Critical points: $(1, 1)$ and $(-1, 1)$ .
Second-order partial derivatives:
$f_{xx} = 12x, \quad f_{yy} = 4, \quad f_{xy} = 0$
Discriminant formula:
$D(x, y) = (12x)(4) - (0)^2 = 48x$
Classification:
- At $(1, 1)$ : $D(1, 1) = 48(1) = 48 > 0, \quad f_{xx}(1, 1) = 12(1) = 12 > 0$ Since $D > 0$ and $f_{xx} > 0$ , $(1, 1)$ is a local minimum with value: $f(1, 1) = 2(1)^3 - 6(1) + 2(1)^2 - 4(1) = 2 - 6 + 2 - 4 = -6$
- At $(-1, 1)$ : $D(-1, 1) = 48(-1) = -48 < 0$ Since $D < 0$ , $(-1, 1)$ is a saddle point with value: $f(-1, 1) = 2(-1)^3 - 6(-1) + 2(1)^2 - 4(1) = -2 + 6 + 2 - 4 = 2$

Example 2: Non-Zero Cross-Derivative & Local Maximum