MATLAB Program for Finding Extreme Values with Practical Examples

This document demonstrates how to implement MATLAB programs for calculating extreme values of multivariate functions based on mathematical principles from advanced calculus. We present three practical examples sourced from Tongji University's "Advanced Mathematics" textbook, each illustrating different aspects of extreme value computation with detailed code implementation approaches. The first example focuses on finding extreme values for multivariate functions using MATLAB. We discuss key parameters required for the program and demonstrate how to execute the code with mathematical function inputs. This example introduces MATLAB's fminunc function for unconstrained optimization, explaining how to define objective functions and set initial points for gradient-based search algorithms. The second example extends to ternary functions (three-variable functions), showcasing MATLAB's optimization capabilities for higher-dimensional problems. We detail the implementation using symbolic math toolbox for partial derivative calculations and optimization functions like fminsearch, explaining how to handle multi-variable input parameters and convergence criteria configuration. The third example addresses constrained optimization problems for multivariate functions. We demonstrate constraint implementation using MATLAB's fmincon function, explaining how to define linear/nonlinear constraints through constraint matrices and function handles. This includes discussion of Lagrange multiplier methods and penalty function approaches for handling boundary conditions in optimization algorithms. Each example provides practical insights into translating mathematical problems into executable MATLAB code, emphasizing algorithm selection, parameter tuning, and result validation techniques suitable for engineering and scientific applications.