数学专业外文翻译多元函数的极值.doc

数学专业外文翻译--多元函数的极值 EXTREME VALUES OF FUNCTIONS OF SEVERAL REAL VARIABLES 1. Stationary Points Definition 1.1 Let andThe point a is said to be: 1 a local imum iffor all points sufficiently close to ; 2 a local minimum iffor all points sufficiently close to ; 3 a global or absolute imum 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 imum or minimum. Definition 1.2 Let andThe point a is said to be critical or stationary point if and a singular point if does not exist atFact 1.3 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 ofFact 1.4 If is a continuous function on a closed bounded set then is bounded and attains its bounds. Definition 1.5 A critical point which is neither a local imum nor minimum is called a saddle point. Fact 1.6 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 1.7 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 matrix where 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 matrix where the derivatives are evaluated at. Definition 1.8 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 1.9The Leading Minor Test. Suppose that is a sufficiently smooth function of two variables with a critical point atand H the Hessian of at.If , then is: 1 a local imum if 0detH1 fxx and 0detH; 2 a local minimum if 0detH1 fxx and 0detH; 3 a saddle point if neither of the above hold. where the partial derivatives are evaluated at. Suppose that