多项式函数的极值点与拐点判别及个数公式

本文讨论了多项式函数(x-a)~n,(x-a)~ng(x)(n∈N,n1,a∈R)和∏ki=1(x-a_i)~(n_i)(n_i∈N,n_i0,a_i∈R)的极值点和拐点,并给出了函数∏ki=1(x-a~i)~(n_i)(n_i∈N,n_i0,a_i∈R)所有极值点和拐点的个数公式.     

一道极值点偏移问题的证法分析

<正>设函数f(x)满足f(a)=f(b),并在区间(a,b)内只有一个极值点x_0;若x_0<(a+b)/2,则称极值点x0左偏;若x_0>(a+b)/2,则称极值点x0_右偏.函数f(x)的极值点左偏和右偏统称为函数f(x)的极值点偏移.极值点偏移问题近几年备受命题者的青睐,所涉及思想方法多、思维跨度大、问题变化多端等特点.下面笔者给出一道极值点偏移问题的几种证法,期望读者能举一反三,触类旁通.     

n元函数极值的探讨

在高等数学或数学分析的教学中，经常遇到这样的问题：如何将二元函数求极值的方法推广到，n元函数中去。这一问题在教材中很小涉及，本在[1]、[2]基础上，讨论了，n(n≥2)元函数极值点的判别法，确立了判别临界点为极值点的一个充分条件。这些条件与[1]、[2]的方法是等价的，但方法简单。     

多元函数取局部极值的一个充分条件及其几何意义

在现有的高等数学教材中 ,如文献 [1 ],多元函数取局部极值这一部分仅介绍二元函数在驻点处的情况 ,而有的驻点也无法判断是否为极值点。文献 [2 ]给出了多元函数取局部极值的一个充分条件 ,但也仅考虑驻点的情况 ,有的驻点也无法判断是否为极值点。本文提出的方法 ,对驻点和偏导数不存在的点均能判断是否为极值点 ,且对多元函数本身要求不高。对于二元函数 ,此方法有其明显的几何意义。定理　设 f( x1,x2 ,… ,xn)在 P0 ( x01,x02 ,… ,x0n)的邻域 U( P0 ,δ)内连续 ,且在去心邻域 U( P0 ,δ)内有一阶连续的偏导数。若在 P0 ( x01,x02…     

非闭区间连续函数的最大(小)值问题

<正> 关于连续函数的最大值、最小值问题有两种情况是我们所熟悉的,就是闭区间连续函数和非闭区间内连续且只有唯一极值点的函数的最值问题。那么,我们自然要问,在非闭区间内连续而有若干极值点的函数的最大(小)值在什么条件下存在?若存在如何求解呢?本文就有限个极值点(在严格意义下的极值点,下同)的情况给出解决的一般方法,首先证明两个结论。     

关于拉格朗日乘数法的一点注记

建立了多元函数在任意有限多个约束条件下的极值点和拉格朗日函数极值点之间的一一对应关系,从而找到拉格朗日函数的极值点也就找到了多元函数在这些约束条件下的极值点.从另一角度给出了拉格朗日乘数法的证明.     

关于条件极值的一个注記

本文讨论待求极值的函数以隐函数的形式给出,且约束方程中含有待求极值的函数情况下的条件极值问题。设方程 F(x_1,x_2,…x_n,y)=0~1) (1)在所论某邻域內滿足隐函数存在定理的一切条件,它确定着y对于x_i(i=1,…,n)的函数: y=y(x_1,x_2,…x_n);(2)又设方程组 (m相似文献   

基于矩阵初等变换的n元函数极值的快速判别法

给出了 n元函数极值的一个充分条件 ,并结合矩阵的初等变换建立了 n元函数极值的一种快速判别法 ,最后给出了一个例子     

基于n次型正定性的多元函数极值判别法

针对多元函数稳定点处二阶偏导数全为0的情况,提出了有效的极值判别法.定义了广义n维方阵、n次型及其正定性;提出了更具普遍意义的极值充分条件;得到了利用n次型的正定性判断n元函数极值的方法并举例验证了结论的正确性和有效性.     

分段函数极值问题的研究

通过引进单侧极值的概念,给出了极值存在的充分必要条件,并进一步分析了分段函数单侧极值存在的充分条件.借助符号函数,证明了适用于振荡函数极值存在问题的充分必要条件.对于求导比较复杂或导函数在去心左(右)邻域内变号的极值问题,提出了极值存在的一种充分条件.最后,通过一些有代表性的例子说明了这些方法的有效性.     

一个求解条件极值问题的极值点的新方法

利用齐次线性方程组理论,建立了一个求解条件极值问题的极值点的新方法.该方法的优点是:能有效地避免在运用Lagrange乘数法求解条件极值时,因引进了参数而给解方程组带来的困扰.也可以说,对于有些问题我们仅从已知条件入手,不必引进参数就可以直接求得极值点.     

从几何角度给予拉格朗日乘数法新的推导思路

拉格朗日乘数法是求条件极值的重要方法,该文通过数形结合给出定理推导的新路径,相比教材上纯代数推导更直观,体现了"几何意义"的重要性.     

一元函数的极值及其奇异性

针对一元函数的极值理论,为学生介绍这一理论的进一步发展即奇点理论的一些基本概念和理论.指出二者之间的密切联系.通过对平面曲线的高斯映射以及奇点的介绍,向学生展示高斯曲率与函数的二阶导数以及高斯映射的奇点与函数的拐点之间的奇妙关系.并指出它们的几何意义.同时强调在学习高等数学的相关内容时,应该养成思考它们的几何背景及意义...     

Optimization of the Determinant of the Vandermonde Matrix and Related Matrices

The value of the Vandermonde determinant is optimized over various surfaces, including the sphere, ellipsoid and torus. Lagrange multipliers are used to find a system of polynomial equations which give the local extreme points in its solutions. Using Gröbner basis and other techniques the extreme points are given either explicitly or as roots of polynomials in one variable. The behavior of the Vandermonde determinant is also presented visually in some interesting cases.     

Pathfollowing methods in nonlinear optimization III: Lagrange multiplier embedding

This paper deals with Lagrange multiplier methods which are interpreted as pathfollowing methods. We investigate how successful these methods can be for solving “really nonconvex” problems. Singularity theory developed by Jongen-Jonker-Twilt will be used as a successful tool for providing an answer to this question. Certain modifications of the original Lagrange multiplier method extend the possibilities for solving nonlinear optimization problems, but in the worst case we have to find all connected components in the set of all generalized critical points. That is still an open problem. This paper is a continuation of our research with respect to penalty methods (part I) and exact penalty methods (part II).     

关于条件极值充分条件的重新推导和证明

李文学用拉格朗日函数提出求条件极值的充分条件,但他的证明却是错误的.本文不用拉格朗日函数,而是直接通过消去一个变量将条件极值转化成无条件极值,重新推导出充分性条件.推导的过程也是条件极值充分条件的证明过程.     

A set-valued Lagrange theorem based on a process for convex vector programming

ABSTRACT

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.     

On constraint sets of infinite linear programs over ordered fields

This paper provides a characterization of extreme points and extreme directions of the subsets of the space of generalized finite sequences occurring as the constraint sets of semi-infinite or infinite linear programs. The main result is that these sets are generated by (possibly infinitely many) extreme points and extreme directions. All results are valid over arbitrary ordered fields.     

Random problem generation and the computation of efficient extreme points in multiple objective linear programming

This paper looks at the task of computing efficient extreme points in multiple objective linear programming. Vector maximization software is reviewed and the ADBASE solver for computing all efficient extreme points of a multiple objective linear program is described. To create MOLP test problems, models for random problem generation are discussed. In the computational part of the paper, the numbers of efficient extreme points possessed by MOLPs (including multiple objective transportation problems) of different sizes are reported. In addition, the way the utility values of the efficient extreme points might be distributed over the efficient set for different types of utility functions is investigated. Not surprisingly, results show that it should be easier to find good near-optimal solutions with linear utility functions than with, for instance, Tchebycheff types of utility functions.Dedicated to Professor George B. Dantzig on the occasion of his eightieth birthday.     

The interpolation method of Sprague-Karup

The usual interpolation method is that of Lagrange. The disadvantage of the method is that in the given points the derivatives of the interpolating polynomials are not equal one to the other. In the method of Hermite, polynomials of a higher degree are used, whose derivatives in the given points are supposed to be equal to the derivatives of the function at the given points. This means that those derivatives must be known.If those derivatives are not known, then in the given points the derivatives may be replaced by approximative values, e.g. based on the interpolating polynomials of Lagrange. Such a method has been described by T. B. Sprague (1880) and in a simplified form by J. Karup (1898). In this paper the formulae are derived. Both methods are illustrated with an example. Some properties and theorems are stated. Tables to simplify the computational work are given. Subroutines for these interpolation methods will be published in a next article.