Maxima and Minima of Functions of Two Variables

Summary

Local extrema of $f(x,y)$ occur at critical points where $\mathrm{\nabla }f=\mathbf{0}$ (or where $f$ is not differentiable). The second-derivative (Hessian) test classifies many critical points. On a closed bounded region, also check the boundary.

Prerequisites

Partial Derivatives, Higher-Order Derivatives, Clairaut's Theorem

Procedure

1. Solve $f_x=0$ , $f_y=0$ for critical points in the open domain.
2. At each critical point where second partials exist, compute

3. Classify:
   • $D>0$ and $f_{xx}>0$ : local minimum.
   • $D>0$ and $f_{xx}<0$ : local maximum.
   • $D<0$ : saddle point.
   • $D=0$ : test inconclusive.
4. For absolute extrema on a closed bounded region $R$ , also optimize $f$ on $\partial R$ (one-variable calculus or Lagrange constraints).

Conditions / Assumptions

• Second partials continuous near the critical point for the standard Hessian test.
• Clairaut’s theorem ensures $f_{xy}=f_{yx}$ under continuity of mixed partials, so $D$ is well-defined from either order.

Worked Example

Let $f(x,y)=x^2+y^2-2x-4y+1$ .

Critical point: $f_x=2x-2=0$ , $f_y=2y-4=0$ $\Rightarrow$ $(1,2)$ .

Second partials: $f_{xx}=2$ , $f_{yy}=2$ , $f_{xy}=0$ , so $D=4>0$ and $f_{xx}>0$ : local minimum. Value $f(1,2)=-4$ .

Common Mistakes

• Stopping after finding critical points without the Hessian test or boundary analysis.
• Using $D=f_{xx}+f_{yy}$ (trace) instead of the Hessian determinant.
• Claiming a local extremum is global without further argument.

Connections

• Constrained extrema: Lagrange Multipliers
• Second partials: Higher-Order Derivatives

References

The second-derivative test for functions of two variables is in OpenStax Calculus Volume 3.^[1]