目 錄

摘 要 I
Abstract II
第1章 緒 論 1
1.1研究函數(shù)極值的意義 1
1.2極值的概述 1
第2章 一元函數(shù)極值的求解方法 2
2.1 一元函數(shù)極值定義 2
2.2 一元函數(shù)極值的充分必要條件 2
2.2.1 一元函數(shù)極值的必要條件 2
2.2.2 極值的第一充分條件 2
2.2.3 極值的第二充分條件 3
2.2.4 極值的第三充分條件 4
2.3 一元函數(shù)極值的求解方法 4
第3章 二元函數(shù)極值的求解方法 7
3.1 二元函數(shù)極值定義 7
3.2 二元函數(shù)極值的充分必要條件 7
3.2.1 二元函數(shù)極值必要條件 7
3.2.2 二元函數(shù)極值充分條件 8
3.3二元函數(shù)極值的求法 8
3.4條件極值 9
3.4.1 代入法求極值 9
3.4.2 乘數(shù)法求極值 10
第4章 多元函數(shù)極值的求解方法 12
4.1 多元函數(shù)極值（）定義 12
4.2多元函數(shù)極值的充分必要條件 12
4.2.1 梯度 12
4.2.2 矩陣 12
4.2.3 多元函數(shù)極值必要條件 12
4.2.4 多元函數(shù)極值充分條件 13
4.3 多元函數(shù)極值的求法 14
4.3.1多元函數(shù)的無條件極值求解 14
4.4多元函數(shù)的條件極值求解 15
4.4.1 代入法求極值 15
4.4.2 乘數(shù)法求極值 16
4.4.3 矩陣法求極值 19
4.4.4 梯度法求極值 24
4.4.5 二次方程判別式法求極值 26
4.4.6 標(biāo)準(zhǔn)量代換法 27
結(jié) 束 語 29
致 謝 30
參 考 文 獻(xiàn) 31
附 錄 i
附錄一： 外文文獻(xiàn) i
附錄二： 外文譯文 ix
附錄三： 任務(wù)書 xvii
附錄四： 開題報告 xviii


函數(shù)極值的幾種求法

摘 要
函數(shù)的極值問題是數(shù)學(xué)研究中非常重要的問題，是經(jīng)典微積分最成功的應(yīng)用，它不僅在許多實際問題中占有重要地位，同時也是研究函數(shù)性態(tài)的一個重要特征。在工農(nóng)業(yè)生產(chǎn)、經(jīng)濟(jì)管理和經(jīng)濟(jì)核算中，常常要解決在一定條件下怎么使投入最小，產(chǎn)出最多，效益最高等問題。在生活中也經(jīng)常會遇到求利潤最大化、用料最省、效率最高等問題。這些經(jīng)濟(jì)和生活問題通常都可以轉(zhuǎn)化為數(shù)學(xué)中的函數(shù)問題來探討，進(jìn)而轉(zhuǎn)化為求函數(shù)中最大（?。┲档膯栴}，而且函數(shù)的最大值、最小值問題與函數(shù)的極值有密切聯(lián)系。
本文從一元函數(shù)極值的問題進(jìn)行研究，包括一元函數(shù)的極值的定義，一元函數(shù)極值存在的充分必要條件，以及一元函數(shù)的多種求解方法。依次延伸到二元函數(shù)極值的定義，極值存在的充分必要條件和約束條件下二元函數(shù)極值的各種求解方法，比如代入法、拉格朗日乘數(shù)法。最后再逐步推廣到多元函數(shù)（）極值定義、極值存在的充分必要條件和約束條件下多元函數(shù)極值的各種求解方法。在多元函數(shù)極值方面，尤其是條件極值方面，主要研究的函數(shù)極值的解題方法有利用代入法求極值、拉格朗日(Lagrange)乘數(shù)法求極值、通過雅可比(Jacobi) 矩陣法求條件極值、利用梯度法求極值以及通過二次方程判別式符號法和標(biāo)準(zhǔn)量代換法等初等方法來判別函數(shù)的極值問題，本文旨在對函數(shù)極值的解法問題作出系統(tǒng)性歸納總結(jié)。

關(guān)鍵詞：函數(shù)極值；多元函數(shù)；條件極值；拉格朗日乘數(shù)法；梯度法




Several Methods of Solving the Extremum of Functions
Abstract
Extremum of functions is very important in mathematics research. It is one of the most successful application of classical calculus. Not only does it occupy an important place in many practical problems,but also it is an important characteristic of the property of functions. In industrial and agricultural production, management of the economy and the economic accounting, we often have to solve the problems such as how to use the smallest input to make the most efficient output in given conditions. In our daily life, we often encounter many issues such as how to achieve maximum profit, use the minimum materials and the get maximum efficiency. The above problems can be solved by transforming it to functions in maths, further to maximum or minimum value of functions. And the maximum and minimum value of functions have a close relationship with the extremum of functions.
This paper studies on the issue of extreme value of unary function, including the definition of the extremum of unary function, existence condition of the extremum of unary function and various methods of solving unary function, further to the definition of the extremum of the duality function, existence condition of the extremum of duality function and various methods of solving duality function under constraint condition, such as substitution method and Lagrangian multiplier method. At last, I will promote the definition of the extremum of the multivariate function (), existence condition of the extremum of the multivariate function and various methods of solving the multivariate function under constraint condition. In the extremum of multivariate function, especially in the conditional extremum, to get the extremum of the multivariate function, this paper mainly adopts the following ways: substitution method, Lagrangian multiplier, Jacobi matrix, gradient method, quadratic equations discriminant symbol method and standard substitution method etc. This paper aims to make systemic summary of the extremum of functions.

Key Words: the extremum of functions; the multivariate function; the conditional extremum; Lagrangian multiplier method; gradient method
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