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Math101Multivariable Optimization
A rigorous, example-driven guide to multivariable optimization, including hypotheses, method choice, verification, and practice.
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
- Determine the domain and whether boundaries or corners exist.
- Solve all first-derivative critical equations.
- Compute $f_{xx},f_{yy},f_{xy}$ and evaluate $D$ at each stationary point.
- Use the second-derivative classification or another local method when inconclusive.
- For absolute extrema, analyze boundary pieces and compare every candidate value.
