【正文】 本科畢業(yè) 設(shè)計（論文 ） ( 20xx 屆 ) 題 目： 不等式的證明及其運用 專 業(yè)： 數(shù)學(xué)與應(yīng)用數(shù)學(xué) 班 級： 09 數(shù)學(xué)與應(yīng)用數(shù)學(xué) 姓 名： 王乃澤 學(xué) 號： 09205013247 指導(dǎo)教師： 歐建光 職 稱： 副教授 完成日期： 20xx 年 4 月 20 日 1 不等式證明及其運用 王乃澤 （溫州大學(xué) 甌江 學(xué)院，浙江溫州， 325027） 摘要 ：不等式證明及其應(yīng)用在數(shù)學(xué)中有著不可或缺作用和地位，從初等數(shù)學(xué)到高等數(shù)學(xué)，不等式一直同我們形影不離，它的應(yīng)用范圍非常廣泛，是數(shù)學(xué)教學(xué)容重要組成部分。在不等式的證明過程中需要用到諸多的數(shù)學(xué)思想，結(jié)合了許多重要的數(shù)學(xué)內(nèi)容，本篇論文主要介紹幾個著名不等式之 間證明，運用，以及聯(lián)系，幫助大家區(qū)分解決如何合理有效的運用這些不等式來達(dá)到自己所想要的預(yù)期效果。這幾個不等式也是我們經(jīng)常在學(xué)習(xí)中所要用的，具體的來說，就是通過凸函數(shù)的相關(guān)定義及其性質(zhì)，進(jìn)而引入Jensen 不等式，由 Jensen 不等式推導(dǎo)所要的 holder 不等式，從 holder 不等式中我們看出，只要稍加變形就是大家廣為熟知的柯西不等式。而柯西不等式是本篇論文討論的重點內(nèi)容，我們將著重討論柯西不等式的幾種主要表現(xiàn)形式及相關(guān)的證明，應(yīng)用舉例等等。在此之后我們還將通過柯西不等式推導(dǎo)著名的均值不等式，從均值不等式回到 Jensen 不等式的相關(guān)內(nèi)容。至此，為本篇論文所論述的重要內(nèi)容。 關(guān)鍵詞 ： 凸函數(shù) ；不等式； 2 Inequality proof and its application Wangnaize Oujiang College, Wenzhou University， Wenzhou, Zhejiang， 325027 Abstract： Inequality proof and its application in mathematics has a indispensable role and status, from elementary mathematics to higher mathematics, inequality has been and we were like peas and carrots, its application range is very wide, is an important part of the capacity of mathematics teaching. In the inequality proof process need to use many mathematical thought, bined with many important mathematical content, this paper mainly introduces several famous between inequality proof, use, and contact, help you distinguish between solve how to reasonably and effectively use the inequality to achieve their desired expected effect. These a few inequality is we often in the study will use, concrete, it is through the convex function related definition and nature, and then introduce Jensen inequality, Jensen inequality is derived by the holder inequality, from holder inequality we see, as long as everyone is a widely known as the deformation of Cauchy inequality. And Cauchy inequality is discussed in this paper the key content, we will mainly discuss the Cauchy inequality several main forms and relevant proof, examples of application and so on. After that we will through the Cauchy inequality is famous mean inequality, from mean inequality back to Jensen inequality related content. So far, this paper discusses the important content. Keywords: convex functions。 Inequality。 3 目錄 前言 .................................................................................................................................................4 1 凸函數(shù)的性質(zhì)及其應(yīng)用 .......................................................................................................5 .....................................................................................................5 由 Jensen 不等式推導(dǎo) holder 不等式的相關(guān)證明 ...................... 7 2 柯西不等式 ..............................................................................................................................9 （ Cauchy）不等式 ............................................................................................9 柯西不等式幾何證明 .............................................................................................9 柯西不等式的主要形式 ....................................................................................... 10 ................................................... 14 應(yīng)用柯西不等式求最值 ....................................................................................... 14 柯西不等式推導(dǎo)點到直線的距離公式 ............................................................... 15 柯西不等式求解有關(guān)三角形的問題 ............................................................. 15 5 由柯西不等式求解方程組 .................................................................................. 16 由柯西不等式求解有關(guān)三角函數(shù)不等式問題 .................................................. 17 等式推導(dǎo)均值不等式的部分相關(guān)證明 ........................................... 18 3 均值不等式的應(yīng)用 .............................................................................................................. 19 .......................................................................................... 19 由均值不等式推導(dǎo) Jensen 不等式的個例 ...................................................... 22 致謝 ............................................................................................................................................... 23 參考文獻(xiàn) ....................................................................................................................