对柯西不等式的研究.doc
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对柯西不等式的研究,对一个不等式问题的研究----摘要 除了等式关系，非常多的不等式关系在自然界中存在着，对不等式关系的研究在数学发展特别是应用数学中起着非常重要的作用。不等式问题的解决以“方法巧，多入口”为人所熟知，其涵盖面广，综合性很强，是现在各个级别数学竞赛的热门和难点之一。本文先对柯西不等式从定理、推论、变形、推...
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对一个不等式问题的研究----对柯西不等式的研究
摘要
除了等式关系，非常多的不等式关系在自然界中存在着，对不等式关系的研究在数学发展特别是应用数学中起着非常重要的作用。不等式问题的解决以“方法巧，多入口”为人所熟知，其涵盖面广，综合性很强，是现在各个级别数学竞赛的热门和难点之一。本文先对柯西不等式从定理、推论、变形、推广和积分形式等方面进行了总结，列举了柯西不等式的几种形式及常见的证明方法，以及在高考数学和数学竞赛方面的一些应用。
关键词：柯西不等式，数学竞赛，高考数学
The study of an inequality problem
Abstract：Except for equality of relationships, it is also the most fundamental mathematical relationships. Inequality problems have wide coverage. Inequality plays an important role in the mathematical development, especially mathematical application. It has a strong synthesis, which is one of the top and difficult in the each grade of the mathematics competition.This paper first summarizes the Cauchy inequality from the theorem, inference, deformation, promotion and other aspects of the integral form This article lists several forms Cauchy inequality, and were proved them, and then give some exercises by the Cauchy Inequality in the college entrance examination in mathematics and math competition.
Keywords: Cauchy inequality，Mathematics Competition，National College Entrance Examination Math
目录
第1章.柯西不等式的内容
1.1 柯西不等式的几种形式.................................................................................1
1.2 柯西不等式的推论.........................................................................................2
第2章.柯西不等式的证明方法
2.1 N维形式的几种证明方法
2.1.1 凑平方法.................................................................................................3
2.1.2 法.........................................................................................................3
2.1.3 数学归纳法.............................................................................................4
2.1.4 基本不等式法.........................................................................................5
2.1.5 推广不等式法.........................................................................................6
2.1.6 二次型法.................................................................................................6
2.1.7 向量内积法.............................................................................................7
2.2 推广形式（卡尔松不等式）..........................................................................7
2.3 积分形式的证明.............................................................................................8
2.4 概率论形式的证明.........................................................................................8
第3章.柯西不等式的高考应用
3.1 解析几何中的应用.........................................................................................9
3.2 立体几何中的应用.......................................................................................10
3.3 三角函数中的应用.......................................................................................10
3.4 数列中的应用...............................................................................................11
第4章.柯西不等式的竞赛应用
4.1 求最值的应用...............................................................................................12
4.2 证明不等式的应用.......................................................................................14
4.3 组合数学中的应用.......................................................................................20
总结…………………………………………….........................................................22
致谢.............................................................................................................................23
参考文献...................................................................................................................24
摘要
除了等式关系，非常多的不等式关系在自然界中存在着，对不等式关系的研究在数学发展特别是应用数学中起着非常重要的作用。不等式问题的解决以“方法巧，多入口”为人所熟知，其涵盖面广，综合性很强，是现在各个级别数学竞赛的热门和难点之一。本文先对柯西不等式从定理、推论、变形、推广和积分形式等方面进行了总结，列举了柯西不等式的几种形式及常见的证明方法，以及在高考数学和数学竞赛方面的一些应用。
关键词：柯西不等式，数学竞赛，高考数学
The study of an inequality problem
Abstract：Except for equality of relationships, it is also the most fundamental mathematical relationships. Inequality problems have wide coverage. Inequality plays an important role in the mathematical development, especially mathematical application. It has a strong synthesis, which is one of the top and difficult in the each grade of the mathematics competition.This paper first summarizes the Cauchy inequality from the theorem, inference, deformation, promotion and other aspects of the integral form This article lists several forms Cauchy inequality, and were proved them, and then give some exercises by the Cauchy Inequality in the college entrance examination in mathematics and math competition.
Keywords: Cauchy inequality，Mathematics Competition，National College Entrance Examination Math
目录
第1章.柯西不等式的内容
1.1 柯西不等式的几种形式.................................................................................1
1.2 柯西不等式的推论.........................................................................................2
第2章.柯西不等式的证明方法
2.1 N维形式的几种证明方法
2.1.1 凑平方法.................................................................................................3
2.1.2 法.........................................................................................................3
2.1.3 数学归纳法.............................................................................................4
2.1.4 基本不等式法.........................................................................................5
2.1.5 推广不等式法.........................................................................................6
2.1.6 二次型法.................................................................................................6
2.1.7 向量内积法.............................................................................................7
2.2 推广形式（卡尔松不等式）..........................................................................7
2.3 积分形式的证明.............................................................................................8
2.4 概率论形式的证明.........................................................................................8
第3章.柯西不等式的高考应用
3.1 解析几何中的应用.........................................................................................9
3.2 立体几何中的应用.......................................................................................10
3.3 三角函数中的应用.......................................................................................10
3.4 数列中的应用...............................................................................................11
第4章.柯西不等式的竞赛应用
4.1 求最值的应用...............................................................................................12
4.2 证明不等式的应用.......................................................................................14
4.3 组合数学中的应用.......................................................................................20
总结…………………………………………….........................................................22
致谢.............................................................................................................................23
参考文献...................................................................................................................24
