【正文】 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.Note that the equation is equivalent to the equations,and So, in the two variable case, we have Lagranian function and are solving the equations:, , and .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。y,177。(2)如果對于所有充分地接近時，則是一個局部極小值。(4) 如果對于所有點成立，則是一個全局極小值（或絕對極小值）。 (2) 一個局部極小值點， 如果0det(H1) = fxx并且0det(H)=。(2) 一個局部極小值點， 如果當(dāng)0det(H1), 0det(H2) 并且 0det(H3)時。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.穩(wěn)定點可以分為局部極大值點、局部極小值點或鞍點. 確定在限制條件g(x) = 0下的函數(shù)極值: (1) 計算, 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) 計算在這些所有點處的函數(shù)f的值，并找出所需的極值.當(dāng)限制條件大于一個時，我們則需要解決以下方程.定理 使并且是曲線C上的一個點, 有方程 g(x,y) = 0成立，則在限制條件C ，點不是曲線的端點，且. 因此存在的值使得點是Lagrange函數(shù)的關(guān)鍵點.. 因為點不是曲線的端點，且,則曲線在點處的切線與有關(guān). 如果在點處與平行， f 的值隨著在的運(yùn)動增加減小,所以點不是極值點. 因為和平行，所以存在使得成立. 例. 求內(nèi)接于橢球的體積最大的長方體的體積，長方體的各個面平行于坐標(biāo)面 解：明顯地，當(dāng)長方體的體積最大時，長方體的各個頂點一定在橢球上. 設(shè)長方體的一個頂點坐標(biāo)為(x, y, z) (x0, y0, z0), 則長方體的其他頂點坐標(biāo)分別為(177。y,177。三、重點研究問題重點研究多元函數(shù)極值的一些求法。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)計開題報告 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。因此解決這些問題具有現(xiàn)實意義,通常這些經(jīng)濟(jì)和生活問題都可以轉(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}，也即函數(shù)的極值問題。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。二、研究目標(biāo) 文章將從一元函數(shù)極值的問題開始進(jìn)行研究，包括一元函數(shù)中含參量函數(shù)的極值求解方法，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，對各種函數(shù)極值的解題方法進(jìn)行了歸納與總結(jié)，并通過具體實例對各種解法進(jìn)行分析類比，從中可以得出不同的函數(shù)極值問題可以有不同的解題方法，力爭總結(jié)出一些新的解題方法或者對某種算法進(jìn)行適當(dāng)性的改進(jìn)。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。第4周到第5周： 進(jìn)行資料查閱，知識的回顧復(fù)習(xí)，以確定主要努力的方向和目標(biāo)。第13周到第14周：最終完成畢業(yè)論文，進(jìn)行論文答辯。xx