對一個不等式問題的研究----對柯西不等式的研究



摘要
除了等式關(guān)系，非常多的不等式關(guān)系在自然界中存在著，對不等式關(guān)系的研究在數(shù)學(xué)發(fā)展特別是應(yīng)用數(shù)學(xué)中起著非常重要的作用。不等式問題的解決以“方法巧，多入口”為人所熟知，其涵蓋面廣，綜合性很強(qiáng)，是現(xiàn)在各個級別數(shù)學(xué)競賽的熱門和難點(diǎn)之一。本文先對柯西不等式從定理、推論、變形、推廣和積分形式等方面進(jìn)行了總結(jié)，列舉了柯西不等式的幾種形式及常見的證明方法，以及在高考數(shù)學(xué)和數(shù)學(xué)競賽方面的一些應(yīng)用。

關(guān)鍵詞：柯西不等式，數(shù)學(xué)競賽，高考數(shù)學(xué)


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




目錄
第1章.柯西不等式的內(nèi)容
1.1 柯西不等式的幾種形式.................................................................................1
1.2 柯西不等式的推論.........................................................................................2
第2章.柯西不等式的證明方法
2.1 N維形式的幾種證明方法
2.1.1 湊平方法.................................................................................................3
2.1.2 法.........................................................................................................3
2.1.3 數(shù)學(xué)歸納法.............................................................................................4
2.1.4 基本不等式法.........................................................................................5
2.1.5 推廣不等式法.........................................................................................6
2.1.6 二次型法.................................................................................................6
2.1.7 向量內(nèi)積法.............................................................................................7
2.2 推廣形式（卡爾松不等式）..........................................................................7
2.3 積分形式的證明.............................................................................................8
2.4 概率論形式的證明.........................................................................................8
第3章.柯西不等式的高考應(yīng)用
3.1 解析幾何中的應(yīng)用.........................................................................................9
3.2 立體幾何中的應(yīng)用.......................................................................................10
3.3 三角函數(shù)中的應(yīng)用.......................................................................................10
3.4 數(shù)列中的應(yīng)用...............................................................................................11

第4章.柯西不等式的競賽應(yīng)用
4.1 求最值的應(yīng)用...............................................................................................12
4.2 證明不等式的應(yīng)用.......................................................................................14
4.3 組合數(shù)學(xué)中的應(yīng)用.......................................................................................20
總結(jié)…………………………………………….........................................................22
致謝.............................................................................................................................23
參考文獻(xiàn)...................................................................................................................24