Numerical differentiation methods and their applications in solving function extremes

This paper presents a thorough investigation of numerical differentiation techniques and their application in identifying function extrema. It begins by examining the fundamental concepts of function extrema and then reviews three primary numerical differentiation methods: the finite difference method (with a focus on the central difference scheme), the interpolation-based numerical differentiation method, and the spline interpolation differentiation method. The underlying principles and associated errors of these methods are meticulously analyzed. Building on the theoretical foundations of function extrema, the paper explores how these methods can be applied to locate function extrema. Through targeted numerical experiments, the paper evaluates the accuracy and appropriate contexts for each method, providing a robust theoretical framework for selecting numerical differentiation methods in engineering optimization problems.