多元函数的极值及其应用.doc

多元函数的极值及其应用 摘 要 函数极值一直是数学研究的重要内容之一，在科学与生产实践中存在着许多和极值有关问题。在数学分析中，讨论了二元函数的极值、极值的充分、必要条件、条件极值以及求条件极值的拉格朗日乘数法。本论文讨论了多元函数的极值及其应用。在第一章中，我们将二元函数极值的讨论形式上推广到多元函数的极值上，给出了多元函数极值的充分、必要条件。在第二章中，通过具体实例，我们总结讨论了多元函数的条件极值的求法，有代入消元法、拉格朗日乘数法、梯度法、不等式、二次方程判别式、数形结合法等五个方法，从中可以看到不同的条件极值问题可以有不同的求法，同一个极值问题也可以有几种不同的求法，所以我们需选择适当的方法，掌握正确的解题思路。在第三章中，我们总结归纳了多元函数的极值、条件极值在实际生活中的各种应用，在数学上，我们可以用来证明某些不等式、讨论圆内切边形的最大面积等，在经济学中，可用来讨论效益最优化的问题，在物理学上，可用来讨论光的折射的最短路径等等，由此，我们可以体会到如何将实际生活中的问题转化成讨论极值问题的数学模型，并用我们总结的方法予以解决。 关键词：多元函数 极值 条件极值 拉格朗日乘数法 Extreme Value Of Multivariate Function And Its Application ABSTRACT Function extreme value has always been an important content of mathematical research, there are many in the science and production practice related to extreme value problem.In mathematical analysis, discussed the the extremum of the extreme value of binary function, necessary and sufficient conditions, conditional extremum and Lagrange multiplier method of conditional extreme value. The extremum of function of many variables and its application are discussed in this paper.In the first chapter, we will discuss formally promotion to the extreme value of binary function on the extreme value of multivariate function is given, the necessary and sufficient conditions for the the extremum of function of many variables.In the second chapter, through specific examples, we summarize the conditional extreme value of multivariate function is discussed, the application of minimal polyomial to have substitution elimination method, the Lagrange multiplier method, gradient method, inequality, quadratic equation discriminant, in combination with number form, such as the five methods, we can see different conditional extreme value problem can have different religion, there are some different in this extremum problems can also be the same, so we need to choose the appropriate method, to master the correct way.In the third chapter, we summarized the extremum of function of many variables, conditional extreme value in t