多元函数极值问题探究--0932098--李开.docx

中央民族大学学士论文Bachelor Thesis of Minzu University of China 多元函数极值问题探究An Extremum Exploration of Multivariate Function姓名:李开 学号:0932098年级:09级 院系:理学院 专业:信息与计算科学 指导老师:李成岳 日期:2013/4/24 摘要本文首先介绍了二元函数极值的定义，并运用二元函数的泰勒公式和连续性定理证明了二元函数取极值的必要条件和充分条件，着重讨论了临界条件下判别式等于零的情况，并给出了进一步讨论的方法，之后利用曲面理论引进了二元函数极值问题的几何意义并结合坐标平移法给出了求一些无稳定点的二元函数的极值的方法。本文接着将二元函数推广至多元函数，又结合高等代数中二次型理论及微分几何中曲面第二基本形式理论给出了多元函数极值的定义，必要条件，充分条件和几何意义并予以了证明。本文还介绍了多元函数条件极值的定义，必要条件和充分条件，并引入了拉格朗日乘数法这一求条件极值的有力工具。本文最后给出了多元函数极值理论的一些应用，如最小二乘法，空间距离和不等式的证明以及在实际运用多元函数极值理论求解时的一些注意事项和技巧策略。关键词：泰勒公式 二次型 曲面基本形式 拉格朗日乘数法Abstract Firstly this thesis introduces the definition of binary function extremum, proves the necessary and sufficient conditions of the binary function at its extremal point using Taylor’s formula and continuity theorem, offers a method of further exploration on critical condition where the discriminant equals to zero, explains the geometrical meaning of binary function extremum based on curved surface theory, and puts forward a method of seeking the extremal point of some binary functions without stationary points using coordinate translation. Secondly this thesis extends binary function extremum to multivariate, introduces its definition, proves its necessary and sufficient conditions at its extremal point, and explains its geometrical meaning combined with the quadratic form theory of advanced algebra and the second fundamental form theory of curved surface of differential geometry. Besides this thesis describes the definition of multivariate function conditional extremum, and introduces Lagrange multiplier method, a useful tool solving conditional extremum problems, when proving its necessary and sufficient conditions. Finally this thesis introduces some applications of multivariate function extremum theory, such as least square method, problems concerning spatial distance and inequality proof, and some precautions and strategies when solving problems involved using multivariate function extremum the