【文章內(nèi)容簡介】 本文通過從一元函數(shù)極值的問題開始進行研究，包括一元函數(shù)的極值多種求解方法，其次為二元函數(shù)的常用求解，再逐步推廣到多元函數(shù)極值的各種求解方法。通過對多元函數(shù)條件極值的各種解法及應用的介紹，我們知道對于不同的多元函數(shù)其極值有不同的解法，針對不同的題目要求，我們應該選擇一種既簡便易行又節(jié)省時間的方法，其中拉格朗日乘數(shù)法是一種通用的方法，也是最常用的方法。通過本文知道，除了拉格朗日乘數(shù)法、雅可比矩陣法和梯度法外，其余條件極值解法均為初等數(shù)學的方法，掌握好初等數(shù)學的方法求解多元函數(shù)條件極值有時候會更簡單，但其使用的過程中具有一定的技巧性，也有一定的局限性，需要根據(jù)具體情況具體分析。 當然，僅僅一個學期的論文設計，不足之處在所難免，還希望各位老師指正批評。致 謝四年的讀書生活在這個季節(jié)即將劃上一個句號，而于我的人生卻只是一個逗號，我將面對又一次征程的開始。四年的求學生涯在師長、親友的大力支持下，走得辛苦卻也收獲滿囊，在論文即將付梓之際，思緒萬千，內(nèi)心充滿了無限的感激之情。本次學位論文是在我的導師楊建偉老師的親切關(guān)懷和悉心指導下完成的。楊老師嚴肅的科學態(tài)度，嚴謹?shù)闹螌W精神，使我受益匪淺。在此我要向楊老師表示我最誠摯的謝意和最崇高的敬意！同時也要感謝參考文獻中的作者們，因為你們的貢獻，讓我順利的完成我的論文。感謝母校為我提供的良好學習環(huán)境，使我能夠在此專心學習，陶冶情操。感謝王婧老師在四年大學生活中對我的照顧與關(guān)心，感謝祁萌書記和趙娟老師對我平時的指導以及對我畢業(yè)擇業(yè)時的建議，同時也要感謝陪我一起走過大學四年的同學與朋友，因為你們，我在大學四年經(jīng)歷了許許多多的學生工作經(jīng)歷，讓我受益很多。最后，我更要感謝我的父母，感謝他們對我的養(yǎng)育之恩，更感謝他們對我學業(yè)的支持與默默奉獻。最后我要用我最真誠的心意說聲：“感謝你們！”參 考 文 獻[1][J].考試周刊,2011,(52):8485.[2](第三版)上冊[M].高等教育出版社,.[3][J].邢臺學院學報,:9798.[4][J].武漢交通管理干部學院學報,:110115.[5][J].科技信息,2010,(24):120.[6]“陷阱”[J].數(shù)學技術(shù)應用科學,2006:1516.[7][J].焦作師范高等?？茖W校學報,2007(12):8082.[8][J].氣象教育與科技,2008,3.(2),1418.[9]龍莉,[J].鞍山師范學院學報,5(4):1012.[10][J].菏澤師專學報,25(2):1618. [11]張芳,[J].大學數(shù)學,24(6):130133.[12]齊新社,包敬民,[J]. 高等數(shù)學研究,12(2):5456.[13][D]:玉林師范學院,2007.[14][J].淮南師范學院學報,2004,3(6):12.[15]朱江紅,[J].滄州師范?？茖W校學報,2010,02:9599.[16][J].科技創(chuàng)新導報,2008,(15):246247.[17][J].臨沂師范學院學報,1999,(12):2124.附 錄附錄一：外文文獻EXTREME VALUES OF FUNCTIONS OF SEVERAL REAL VARIABLES1. Stationary PointsDefinition Let and . The point a is said to be: (1) a local maximum iffor all points sufficiently close to 。(2) a local minimum iffor all points sufficiently close to 。(3) a global (or absolute) maximum iffor all points 。(4) a global (or absolute) minimum iffor all points 。(5) a local or global extremum if it is a local or global maximum or minimum.Definition Let and . The point a is said to be critical or stationary point if and a singular point if does not exist at .Fact Let and .If has a local or global extremum at the point , then must be either:(1) a critical point of , or(2) a singular point of , or(3) a boundary point of .Fact If is a continuous function on a closed bounded set then is bounded and attains its bounds.Definition A critical point which is neither a local maximum nor minimum is called a saddle point.Fact A critical point is a saddle point if and only if there are arbitrarily small values of for which takes both positive and negative values.Definition If is a function of two variables such that all second order partial derivatives exist at the point , then the Hessian matrix of at is the matrixwhere the derivatives are evaluated at.If is a function of three variables such that all second order partial derivatives exist at the point , then the Hessian of f at is the matrixwhere the derivatives are evaluated at.Definition Let be an matrix and, for each ,let be the matrix formed from the first rows and columns of .The determinants det(),are called the leading minors of Theorem (The Leading Minor Test). Suppose that is a sufficiently smooth function of two variables with a critical point atand H the Hessian of , then is:(1) a local maximum if 0det(H1) = fxx and 0det(H)=。 (2) a local minimum if 0det(H1) = fxx and 0det(H)=。 (3) a saddle point if neither of the above hold.where the partial derivatives are evaluated at.Suppose that is a sufficiently smooth function of three variables with a critical point at and Hessian H , then is:(1) a local maximum if 0det(H1), 0det(H2) and 0det(H3)。(2) a local minimum if 0det(H1), 0det(H2) and 0det(H3)。(3) a saddle point if neither of the above hold.where the partial derivatives are evaluated at.Key Points.A continuous function on a closed bounded set is bounded and achieves its bounds.To find the extreme values of a function on a closed bounded set it is necessary to consider the value of the function at stationary points(), singular points (does not exist) and boundary points(points on the edge of the set).Stationary points can be classified as local maxim , local minim or saddle points.If The Leading Minor Test is not applicable, the stationary point must be classified by directly applying Definition and Fact . For example in the two variable case, if has a stationary point at ,we consider the sign offor arbitrarily small, positive and negative values of and (that are not both zero). In each case, if det(H)= 0, then can be either a local extremum or a saddle point.Example. Find and classify the stationary points of the following f