Multivariable Optimization

The central idea

For differentiable $f(x,y)$, an interior local extremum must occur at a critical point where $f_x=f_y=0$ or the derivative fails. At a stationary point, let $D=f_{xx}f_{yy}-f_{xy}^2$. If $D>0$ and $f_{xx}>0$ there is a local minimum; if $D>0$ and $f_{xx}<0$ a local maximum; if $D<0$ a saddle; $D=0$ is inconclusive.

Definitions, hypotheses, and notation

On a closed bounded region, the Extreme Value Theorem guarantees absolute extrema for continuous $f$, but the boundary can contain its own one-variable critical points and corners. A circular boundary may be handled by parametrization or Lagrange multipliers; a rectangular boundary requires four edge problems plus corners.

The discriminant formula assumes equality of mixed partials near the point, commonly ensured by continuous second partial derivatives. More generally, eigenvalues of the symmetric Hessian determine definiteness. The $2\times2$ test is a compact version of that linear-algebra criterion.

Conceptual meaning

The Hessian quadratic form describes second-order bending in all directions. Positive definite bending creates a bowl, negative definite a cap, and mixed signs a saddle. Critical-point classification is local; absolute extrema require domain and boundary analysis.

A dependable method and decision rule

1. Determine the domain and whether boundaries or corners exist.
2. Solve all first-derivative critical equations.
3. Compute $f_{xx},f_{yy},f_{xy}$ and evaluate $D$ at each stationary point.
4. Use the second-derivative classification or another local method when inconclusive.
5. For absolute extrema, analyze boundary pieces and compare every candidate value.

Fully worked example

Graphical or geometric meaning

Elliptical level curves surround the minimum and increase outward. At a saddle, level curves form hyperbola-like crossings, showing increase in some directions and decrease in others.

Common mistakes and why they fail

Verification and reasonableness checks

• Substitute the critical point into both first partial derivatives.
• Test a few directional slices through a claimed saddle or extremum.
• Compare values on all boundary and interior candidate sets for global questions.

Interior and boundary require different searches

For an unconstrained interior extremum of a differentiable function, solve $\nabla f=\mathbf0$ and include points where derivatives fail. The Hessian test can classify nondegenerate critical points, but determinant zero is inconclusive. On a closed bounded region, continuity guarantees absolute extrema, yet interior analysis alone is insufficient: optimize on every boundary curve and compare all candidate values, including corners and endpoints. A constraint may be handled by substitution or Lagrange multipliers, with singular constraint points checked separately. Keep local classification distinct from the global value comparison. A sketch of level curves and the feasible set often predicts which boundaries matter and exposes missing candidates before algebra becomes elaborate.

Practice

1. Classify $x^2+y^2$ at the origin.
2. Classify $x^2-y^2$ at the origin.
3. What does $D=0$ imply?

Connections and next steps

• Lagrange Multipliers
• Partial Derivatives
• Gradient