泰勒公式外文翻译.doc

PAGE PAGE 11 Taylors Formula and the Study of Extrema Taylors Formula for Mappings Theorem 1. If a mapping from a neighborhood of a point x in a normed space X into a normed space Y has derivatives up to order n -1 inclusive in U and has an n-th order derivative at the point x, then (1) as. Equality (1) is one of the varieties of Taylors formula, written here for rather general classes of mappings. Proof. We prove Taylors formula by induction. For it is true by definition of . Assume formula (1) is true for some . Then by the mean-value theorem, formula (12) of Sect. 10.5, and the induction hypothesis, we obtain. as. We shall not take the time here to discuss other versions of Taylors formula, which are sometimes quite useful. They were discussed earlier in detail for numerical functions. At this point we leave it to the reader to derive them (see, for example, Problem 1 below). Methods of Studying Interior Extrema Using Taylors formula, we shall exhibit necessary conditions and also sufficient conditions for an interior local extremum of real-valued functions defined on an open subset of a normed space. As we shall see, these conditions are analogous to the differential conditions already known to us for an extremum of a real-valued function of a real variable. Theorem 2. Let be a real-valued function defined on an open set U in a normed space X and having continuous derivatives up to order inclusive in a neighborhood of a point and a derivative of order k at the point x itself. If and , then for x to be an extremum of the function f it is: necessary that k be even and that the form be semidefinite, and sufficient that the values of the form on the unit sphere be bounded away from zero; moreover, x is a local minimum if the inequalities , hold on that sphere, and a local maximum if , Proof. For the proof we consider the Taylor expansion (1) of f in a neighborhood of x. The assumptions enable us to write where is a real