Second Derivative Test for Multivariable Local Extrema

Definition

The second derivative test classifies critical points $(a, b)$ of a function of two variables where $f_x = f_y = 0$ using the discriminant (Hessian determinant).

Why It Matters

This test is the ‘multivariable compass’ for finding optimal points; in complex systems with many inputs, it is the only reliable way to know if you are standing on a peak, a valley, or a deceptive saddle point.

Core Concepts

• The Discriminant ($D$): $D = f_{xx} f_{yy} - (f_{xy})^2$

• How to read: “The quantity D equals f x x times f y y minus f x y squared.”

  • Meaning: The determinant of the Hessian matrix—product of pure second partials minus squared mixed partial squared.

• Classification:

  1. Local Min: $D > 0$ and $f_{xx} > 0$.
  2. Local Max: $D > 0$ and $f_{xx} < 0$.
  3. Saddle Point: $D < 0$.
  4. Inconclusive: $D = 0$.

• How to read: “The D positive and f-x-x positive means local minimum; D positive and f-x-x negative means local maximum; D negative means saddle; D zero is inconclusive.”

  • Meaning / when to use: For multivariable critical points where both first partials vanish. $D > 0$ means definite curvature (bowl); $D < 0$ means opposite curvatures in different directions (saddle).

Connected Concepts

• The Second Derivative Test for Local Extrema
• The First Derivative Test for Local Extrema
• The Second Derivative Test for Concavity