【正文】 the Hessian of , then is:(1) a local maximum if 0det(H1) = fxx and 0det(H)=。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 functions: (1) (2) Solution. (1) ,soijkCritical points occur when ,. when(1) (2) (3) Using equations (2) and (3) to eliminate y and z from (1), we see thator ,giving , and .Hence we have three stationary points: , and . Since, and ,the Hessian matrix is At ,which has leading minors 0,And det .By the Leading Minor Test, then, is a local minimum. At ,which has leading minors 0,And det .By the Leading Minor Test, then, is also a local minimum.At , the Hessian isSince det, we can apply the leading minor test which tells us that this is a saddle point since the first leading minor is 0. An alternative method is as follows. In this case we consider the value of the expression，for arbitrarily small values of h, k and l. But for very small h, k and l, cubic terms and above are negligible in parison to quadratic and linear terms, so h, k and l are all positive, . However, if and and ,then .Hence close to ,both increases and decreases, so is a saddle point.(2) soij.Stationary points occur when ,. at .Let us classify this stationary point without considering the Leading Minor Test (in this case the Hessian has determinant 0 at so the test is not applicable).LetCompleting the square we see that So for any arbitrarily small values of h and k, that are not both 0, and we see that f has a local maximum at .2. Constrained Extrema and Lagrange MultipliersDefinition Let f and g be functions of n variables. An extreme value of f(x) subject to the condition g(x) = 0, is called a constrained extreme value and g(x) = 0 is called the constraint.Definition If is a function of n variables, the Lagrangian function of f subject to the constraint is the function of n+1 variableswhere is known as the Lagrange multiplier. The Lagrangian function of f subject to the k constraints , is the function with k Lagrange multipliers, Key Points.(2)如果對于所有充分地接近時，則是一個局部極小值。在有界閉集上的連續(xù)函數(shù)有邊界，并且可以取到其邊界值.y,177。目前國內(nèi)函數(shù)極值主要的求解方法有代入法、拉格朗日乘數(shù)法、標(biāo)準(zhǔn)量代換法、不等式法、二次方程判別式符號法、梯度法、數(shù)形結(jié)合法等等方法。第13周到第14周：最終完成畢業(yè)論文，進(jìn)行論文答辯。采取的主要技術(shù)路線或方法 參考大量的相關(guān)的書籍以及相關(guān)論文，通過如中國學(xué)術(shù)期刊網(wǎng)、萬方數(shù)據(jù)資源系統(tǒng)、中國知網(wǎng)等中文數(shù)據(jù)庫及外文數(shù)據(jù)庫的檢索收集資料； 借助所學(xué)數(shù)學(xué)知識和理論，尤其是數(shù)學(xué)分析方面極限理論、微積分理論，深入分析題目，提出提綱，確定論文思路； 整理各種函數(shù)極值（尤其是多元函數(shù)極值、函數(shù)的無條件極值和約束條件極值）求解方法，對求解函數(shù)極值的求解方法做一總結(jié)； 對各種方法進(jìn)行對比，分析，并對函數(shù)極值的求法進(jìn)行深入；預(yù)期的成果及形式通過對求函數(shù)極值的不同求解方法進(jìn)行深入探研，并最終形成15000字的論文。選題來源：自選 指導(dǎo)教師： 年 月 日附錄四：華北水利水電學(xué)院本科生畢業(yè)設(shè)計(jì)開題報(bào)告 2012年3月7日學(xué)生姓名 學(xué)號 專業(yè)數(shù)學(xué)與應(yīng)用數(shù)學(xué)題目名稱函數(shù)極值的幾種求法課題來源自 選選題依據(jù)一、研究的意義函數(shù)的極值一直是數(shù)學(xué)研究的重要內(nèi)容之一，由于它的應(yīng)用廣泛，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。確定在限制條件g(x) = 0下的函數(shù)極值: (1) 計(jì)算, 并且記住其是一個含有n+1個變量（和）的函數(shù) (2) 得出使的值 (不需要明確的算出相對應(yīng)的的值): (3) 計(jì)算在這些所有點(diǎn)處的函數(shù)f的值，并找出所需的極值. (2) 一個局部極小值點(diǎn)， 如果0det(H1) = fxx并且0det(H)=。such that and the result follows.Example. Find the rectangular box with the largest volume that fits inside the ellipsoid ,given that it sides are parallel to the axes.Solution. Clearly the box will have the greatest volume if each of its corners touch the ellipse. Let one corner of the box be corner (x, y, z) in the positive octant, then the box has corners (177。(3) a saddle point if neither of the above hold.where the partial derivatives are evaluated at.Key Points.最后我要用我最真誠的心意說聲：“感謝你們！”參 考 文 獻(xiàn)[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(xué)院學(xué)報(bào),:9798.[4][J].武漢交通管理干部學(xué)院學(xué)報(bào),:110115.[5][J].科技信息,2010,(24):120.[6]“陷阱”[J].數(shù)學(xué)技術(shù)應(yīng)用科學(xué),2006:1516.[7][J].焦作師范高等?？茖W(xué)校學(xué)報(bào),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(xué)院學(xué)報(bào),5(4):1012.[10][J].菏澤師專學(xué)報(bào),25(2):1618. [11]張芳,[J].大學(xué)數(shù)學(xué),24(6):130133.[12]齊新社,包敬民,[J]. 高等數(shù)學(xué)研究,12(2):5456.[13][D]:玉林師范學(xué)院,2007.[14][J].淮南師范學(xué)院學(xué)報(bào),2004,3(6):12.[15]朱江紅,[J].滄州師范?？茖W(xué)校學(xué)報(bào),2010,02:9599.[16][J].科技創(chuàng)新導(dǎo)報(bào),2008,(15):246247.[17][J].臨沂師范學(xué)院學(xué)報(bào),1999,(12):2124.附 錄附錄一：外文文獻(xiàn)EXTREME VALUES OF FUNCTIONS OF SEVERAL REAL VARIABLES1. Stationary PointsDefinitio