求解多元函数最值的策略

Strategies for Solving the Maximum Value of Multivariate Functions

归结了求解多元函数最值问题的若干策略与方法,如消元策略、换元策略、主元策略、引元策略、构造策略、数形结合策略、对称策略等.其中消元策略包括代入消元、增量消元、凑配消元、放缩消元等方法;换元策略包括三角换元、整体换元等方法;引元策略包括引入一个参数、两个参数、多个参数等方法;不等式策略包括均值不等式、柯西不等式、椭圆不等式、降幂不等式、权方和不等式等方法;构造策略包括构造函数、构造复数、构造对偶式、构造向量等方法;应用数形结合策略和对称策略解决多元函数最值问题一般都比较简洁.同时运用这些策略和方法分析并解决了一些典型问题.多元函数最值问题的教学是实现"授人以渔"的良好路径.

Some strategies for solving the extreme value problems related to multivariate functions were summed up,including the elimination strategy,commutation strategy,principal element strategy,variables introduction strategy,construction strategy,combination strategy of number and form,symmetry strategy,etc.,of which the elimination strategies include substitutional elimination,incremental elimination,matching elimination,reduction elimination and so forth;The substitution strategy includes trigonometric substitution,integral substitution and so on.The strategy related to the introduction of variables includes the introduction of one parameter,two parameters as well as multiple parameters.Inequality strategy contains methods like mean inequality,Cauchy inequality,elliptic inequality,power reduction inequality,weighting inequalities,etc.Construction strategy includes constructed function,constructed complex numbers,constructed duality and constructed vectors and so on.The application of the combined strategy of numbers and shapes together with the symmetric strategies in solving the extreme value problems related to multivariate functions is found to be quick and concise.These strategies and methods have been used to analyze and solve some typical problems.