Extrema of Multivariable Functions: Local, Global, and Constrained Optimization
DOI:
https://doi.org/10.69760/portuni.26060005Keywords:
Multivariable functions, extrema, local maximum, local minimum, global extremum, critical points, Hessian matrix, Lagrange multipliers, constrained optimization, second-order conditionsAbstract
This article presents a rigorous and comprehensive exposition of the theory of extrema for multivariable functions, covering the foundational definitions, necessary and sufficient conditions for local extrema, the distinction between local and global (absolute) extrema on closed bounded domains, and the method of Lagrange multipliers for constrained optimization. Beginning from precise definitions of maxima and minima of two-variable functions, the article develops the second-order sufficient conditions via the discriminant of the Hessian matrix, extends these conditions to three-variable functions using principal minors, and establishes the procedure for finding global extrema on closed domains. The theory of constrained extrema is developed through the construction of the Lagrange function, the formulation of necessary conditions as a stationary system, and the application of the bordered Hessian determinant as a sufficient condition. All theoretical results are illustrated with explicit worked examples, including the determination of the global extremum of z = x² + y² on a rectangular domain and the constrained extremum of z = xy subject to the linear constraint 2x + 3y = 5. The article aims to provide students and researchers in mathematics and its applications with a clear, self-contained reference for the theory and practice of multivariable optimization.
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