多元函数的极值与最值的求法【毕业论文】.doc

l l 玉林师范学生院本科生毕业论文（设计） PAGE 2 多元函数的极值与最值的求法 摘要 在实际问题中, 往往会遇到多元函数的最大值、最小值问题.多元函数的最大值、最小值问题与极大值、极小值有密切联系. 求多元函数极值, 一般可以利用偏导数来解决.与一元函数相类似, 可以利用函数的极值来求函数的最大值和最小值，但是由于自变量个数的增加, 从而使该问题更具复杂性. 这里主要讨论二元函数, 对于二元以上的函数极值可以类似加以解决. 求多元函数的极值,本文主要采用以下方法：（1）利用二元函数的偏导数求二元函数极值；（2）拉格朗日乘数法求极值；（3）用几何模型法求解极值;（4）通过Jacobi 矩阵求条件极值；(5)利用参数方程求极值;(6)利用方向导数判别多元函数的极值;(7)用梯度法求极值. 对多元函数的最值问题,我们主要采用的方法有：（1）消元法；（2）均值不等式法；（3）换元法；（4）数形结合法；（5）柯西不等式法；（6）向量法.除此之外，很重要的一种就是：考虑极值与最值的关系，运用极值法求最值. 关键词:多元函数，极值，最值，方法 、 Methods for Calculating Extremum and the most Value of Multivariable Function Author：Chenlong Class: 2007-2 Mathematics and Applied Mathematics Supervisor: Huang Junhua Abstract In practical problems, we often encounter maximum and minimum problems of multivariable function. Both of them have a close relationship with maximum, minimum values. Similar to monad function, we can use the extremum of Function to seek the maximum and minimum value of function, but due to the increased number of independent variable which make the issue more complicated. Usually, we can use the partial derivatives to get the extremum of multivariable function. Here, the thesis mainly discusses the duality function so that we can use the similar way to solve the extremum of duality function to the above. To get the extreme of multivariable function, the thesis adopts the following ways: (1)Using the partial derivative of duality function to get the extreme; (2)Lagrangian multiplier method to calculate the extremum; (3)Geometric modeling method for solving extremum; (4) Using Jacobi matrix to get the conditional extremum; (5) Using parameter equation to calculate the extremum; (6)Using directional derivative to identify the extremum of multivariable function; (7) Using gradient method to get the extremum. To calculate the most value of multivariable function, the thesis takes several main ways as follow: (1) Elimination method (2) The mean value inequality method (3) Substitution method (4) Method