《几何算数平均不等式.docVIP

几何——算术平均不等式及其在代数中的应用 摘 要 。 。 关键词 ；； Geometry - the arithmetic mean inequality and its application in algebra Abstract Geometry - the arithmetic average of inequality is very important inequality，The most widely used in modern analytical mathematics，Many of the conclusions proved to be using this inequality on the basis of，Clever use of this inequality can make many of the problems is a beautiful solution，Brought a lot of convenience for our research work. The proof of this inequality and we are interested in. With the inequality continues to be proven and be used to prove the other conclusions，Lead to the use of inequality greatly advance. Geometry - the arithmetic average of the inequality in the extreme value, the conditional extremum seeking some iterative series limit, series convergence and inequality derivation of a large number of widely used，Apply this inequality can be many unexpected results，It also results of the use and development of a variety of transformation. On the geometry - the arithmetic mean inequality research and extension, our problem-solving ideas will be to develop mathematical thinking will be a corresponding increase in, which is of practical significance to explore some of the substantive issues. Key words Geometry - the arithmetic average of inequality ; Elementary Proof ;The use of inequality 1 引言 1.1 研究背景和意义 。1.2 研究现状 1.3 本文的主要工作及内容 本文。 2 这一节简要介绍2.1不等式的定义和性质 用不等号可以将两个解析式连接起来所成的式子。在一个式子中的数的关系,不全是等号,含不等符号的式子，那它就是一个不等式。例如2x+2y≥2xy，sinx≤1，ex0 ，2x3，5x≠5等 。根据解析式的分类也可对不等式分类，不等号两边的解析式都是代数式的不等式，称为代数不等式；也分一次或多次不等式。只要有一边是超越式，就称为超越不等式。例如lg(1+x)x是超越不等式。 2.2 几何——算术平均不等式的定义 (1)对于任意的n个数x1,x2,x3… xn，这n个数的算术平均值为。 (2)若x1,x2,x3… xn是任意n个非负的实数，这n个数的几何平均值为 。 (3)所谓的几何——算术平均不等式是：;即若 x1,x2,x3… xn是任意n个非负的实数时，其算术平均值必大于或者等于其几何平均值，等号成立的条件为：。 2.3 二元几何——算术平均不等式的基本理论与研究 人民教育出版社出版的全日制普通高级中学教科书教学说明，如果a，b是非负数，那么，当且仅当a=b时取“=”号。即两个非负数的算术平均数不小于他们的几何平均数。这个不等式，我们通常把它称为均值不等式即二元几何——算术平均不等式。对次不等式的深刻理解和掌握。弄清楚其运用条件，便能在解题中快速找到突破口，进而找到正确解决问题的方法。 对二元几何——算术