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1.
题目(2011年山东省高考数学模拟第12题):设函数f(x)的定义域为D,若f(x)满足下面两个条件,则称f(x)为闭函数:①f(x)在D内为单调函数;②存在区间[a,b]∈D,使得f(x)在[a,b]上的值域为[a,b].如果f(x)=√2x+1+k为闭函数,则实数k的取值范围是  相似文献   

2.
曾创 《高等数学研究》2005,8(5):60-60,62
就可微函数的导函数的连续性归纳出一个命题:↓Af(x)∈D(a,b),f′(x)∈C(a,b)<=>f′(x)在(a,b)中不存在第二类间断点。  相似文献   

3.
题 (南京市2009届高三质量检测20)已知函数f(z)=1/2x2-alnx(a∈R), (1)若函数f(x)在x=2处的切线方程为y=x+b,求a,b的值 (2)若函数f(x)在(1,+∞)为增函数,求a的取值范围;  相似文献   

4.
《数学通讯》2008,(4):42-44
题185 已知函数:f(x)=x(1-x)^2. (1)求函数f(x)的极值,并作出函数图象的简图; (2)求实数a,b的值,使在区间[a,b]上的值域也为[a,b]; (3)是否存在区间[a,b](a〈6≤0),使f(x)在区间[a,b]上的值域为[ka,kb],且使k的值最小?若存在求出a,b的值及k的最小值;若不存在,请说明理由.  相似文献   

5.
罗尔定理是说,若f(x)满足:(1)在闭区间[a,b]上连续,(2)在开区间(a,b)内可导,(3)区间端点处的值相等,即f(a)=f(b),则至少存在一点,使得.如果将定理的条件(2)改成f(x)在(a,b)内右导数存在,其它两条不变,是否也存在一点,使得呢?一般不可以.考察函数.显然,(1)f(X)在上连续,切我们有下面定理:定理若函数f(x)在闭区间上连续;在开区间(a,b)内右导数存在且连续(即:存在且连续);且f(a)=f(b),则至少存在一点,使得证明由f(x)在[a,b]上连续,必取到最大值M,最小值m,这样只有两种情形…  相似文献   

6.
从相关习题出发,借助夹逼定理可证明:lim n→∞(b1a^n1+b2a^n2+…+bma6n m)1/n=max{a1,a2,…,am};设函数φ(x),f(x)在[a,b]上都是正连续函数,则有lim n→∞{∫^b aφ(x)[f(x)]^n dx}^1/n=max a≤x≤b{f(x)}  相似文献   

7.
由拉格朗日中值定理很容易得到定理1定理1若函数f(x)在(a,b)内可徽,则对(a,b)内的任意两点x1〈x2,在(x1,x2)内至少有一点ξ(a<ξ<b),使等式成立。那么,若函数x(x)在(a,b)内可微,对于区间(a,b)的内任一点ξ,可否从(a,b)内找到两点X1及x2,满足等式一般不可以。考察函数f(x)=X3,(-1<X<1).对于ξ=0就找不到所需的x1、X2,使(1)成立。事实上,时,但是,当的条件加强时,有定理2定理2若函数f(x)在(a,的内二次可微且产($)一0,a<誊<b,则在区间内可找到两个值由、X。满足f(。)一人X;…  相似文献   

8.
王胜林  方久福 《数学通讯》2010,(9):38-38,40
问题 已知函数f(x)=-+x3+ax2+b(a,b∈R),若函数y=f(x)的图象上任意不同两点连线的斜率小于2,求a的取值范围.  相似文献   

9.
在高三模拟考试中,经常出现下面这类函数题目. 题目 已知函数f(x)=4x-1/x^2+λ/x.若对任意两个不等的正数a,b,有|f(a)-f(b)|〉|a—b|恒成立.求λ的取值范围.  相似文献   

10.
众所周知,闭区间上的连续函数具有介值性。本文要讨论具有介值性的函数的连续性问题,同时还要讨论介值性与原函数的存在性之间的关系。首先指出,在区间[a,b]上具有介值性的函数不必在[a,hi上连续。例如,函数在区间上具有介值性,但却在x=0点不连续。在区间[a,b]上具有介值性的函数在[a,b]上虽然不一定连续,但我们有如下定理:定理1若函数在区间[a,b]上有定义,且在[a,b]上具有介值性,则函数f(x)在区间[a,b]上必不存在跳跃间断点。证用反证法。假设f(x)在区间[a,b]上存在一个跳跃间断点x0,即f(x0-0)、f(x0+0)都…  相似文献   

11.
We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.  相似文献   

12.
We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the t parameter from Hall-Littlewood theory.  相似文献   

13.
张萍萍  李伟年 《数学学报》2018,61(2):243-260
迭代运算下,函数值可以交叉于不同的子区间,使得逐段单调函数的高度异常复杂.本文考虑一个非单调点的连续函数类.首先给出高度的充分必要条件,以此获得此类函数的一种划分.其次针对函数类的一个非空子集,给出判定拓扑共轭的充分必要条件和构造拓扑共轭的新方法.进一步地,我们阐明这样的事实:两个逐段单调函数拓扑共轭是其高度相等的充分不必要条件,最后举例说明.  相似文献   

14.
We give a comprehensive introduction to the algebra of set functions and its generating functions. This algebraic tool allows us to formulate and prove a product theorem for the enumeration of functions of many different kinds, in particular injective functions, surjective functions, matchings and colourings of the vertices of a hypergraph. Moreover, we develop a general duality theory for counting functions.  相似文献   

15.
M. V. Dolgopolik 《Optimization》2017,66(10):1577-1622
In this article, we develop a general theory of exact parametric penalty functions for constrained optimization problems. The main advantage of the method of parametric penalty functions is the fact that a parametric penalty function can be both smooth and exact unlike the standard (i.e. non-parametric) exact penalty functions that are always nonsmooth. We obtain several necessary and/or sufficient conditions for the exactness of parametric penalty functions, and for the zero duality gap property to hold true for these functions. We also prove some convergence results for the method of parametric penalty functions, and derive necessary and sufficient conditions for a parametric penalty function to not have any stationary points outside the set of feasible points of the constrained optimization problem under consideration. In the second part of the paper, we apply the general theory of exact parametric penalty functions to a class of parametric penalty functions introduced by Huyer and Neumaier, and to smoothing approximations of nonsmooth exact penalty functions. The general approach adopted in this article allowed us to unify and significantly sharpen many existing results on parametric penalty functions.  相似文献   

16.
We give a definition of the class of functions with a concave minorant and compare these functions with other classes of functions often used in global optimization, e.g. weakly convex functions, d.c. functions, Lipschitzian functions, continuous and lower semicontinuous functions. It is shown that the class of functions with a concave minorant is closed under operations mainly used in optimization and how a concave minorant can be constructed for a given function.  相似文献   

17.
In this paper, we give some conditions for a class of functions related to Bessel functions to be positive definite or strictly positive definite. We present some properties and relationships involving logarithmically completely monotonic functions and strictly positive definite functions. In particular, we are interested with the modified Bessel functions of the second kind. As applications, we prove the logarithmically monotonicity for a class of functions involving the modified Bessel functions of second kind and we established new inequalities for this function.  相似文献   

18.
The paper is concerned with the filled functions for global optimization of a continuous function of several variables. More general forms of filled functions are presented for smooth and nonsmooth optimizations. These functions have either two adjustable parameters or one adjustable parameter. Conditions on functions and on the values of parameters are given so that the constructed functions are desired filled functions.  相似文献   

19.
The investigation of computational properties of discontinuous functions is an important concern in computable analysis. One method to deal with this subject is to consider effective variants of Borel measurable functions. We introduce such a notion of Borel computability for single‐valued as well as for multi‐valued functions by a direct effectivization of the classical definition. On Baire space the finite levels of the resulting hierarchy of functions can be characterized using a notion of reducibility for functions and corresponding complete functions. We use this classification and an effective version of a Selection Theorem of Bhattacharya‐Srivastava in order to prove a generalization of the Representation Theorem of Kreitz‐Weihrauch for Borel measurable functions on computable metric spaces: such functions are Borel measurable on a certain finite level, if and only if they admit a realizer on Baire space of the same quality. This Representation Theorem enables us to introduce a realizer reducibility for functions on metric spaces and we can extend the completeness result to this reducibility. Besides being very useful by itself, this reducibility leads to a new and effective proof of the Banach‐Hausdorff‐Lebesgue Theorem which connects Borel measurable functions with the Baire functions. Hence, for certain metric spaces the class of Borel computable functions on a certain level is exactly the class of functions which can be expressed as a limit of a pointwise convergent and computable sequence of functions of the next lower level. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)  相似文献   

20.
Near-Subconvexlikeness in Vector Optimization with Set-Valued Functions   总被引:1,自引:0,他引:1  
A new class of generalized convex set-valued functions, termed nearly-subconvexlike functions, is introduced. This class is a generalization of cone-subconvexlike maps, nearly-convexlike set-valued functions, and preinvex set-valued functions. Properties for the nearly-subconvexlike functions are derived and a theorem of the alternative is proved. A Lagrangian multiplier theorem is established and two scalarization theorems are obtained for vector optimization.  相似文献   

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