不等式的证明方法-毕业论文

江西师范大学09届学士学位毕业论文江西师范大学数学与信息科学学院学士学位论文不等式的证明方法Methodtoproveinequality姓名：学号：200907010059学院：数学与信息科学学院专业：数学与应用数学指导老师：完成时间：2013年3月9日江西师范大学09届学士学位毕业论文1不等式的证明方法***【摘要】不等式证明在数学中有着举足轻重的作用和地位，是进行计算、推理、数学思想方法渗透的重要题材，是数学内容的重要组成部分，在不等式的证明过程中需要用到诸多的数学思想，结合了许多重要的数学内容。在本论文中，我总结了一些数学中证明不等式的方法.在初等数学不等式的证明中经常用到的有比较法、作商法、分析法、综合法、数学归纳法、反证法、放缩法、换元法、判别式法、函数法、几何法等等.在高等数学不等式的证明中经常利用中值定理、泰勒公式、拉格朗日函数、以及一些著名不等式，如：均值不等式、柯西不等式、詹森不等式、赫尔德不等式等等.从而使不等式的证明方法更加的完善，有利于我们进一步的探讨和研究不等式的证明.通过学习这些证明方法，可以帮助我们解决一些实际问题，培养逻辑推理论证能力和抽象思维的能力以及养成勤于思考、善于思考的良好学习习惯。【关键词】不等式比较法数学归纳法函数江西师范大学09届学士学位毕业论文2Methodtoproveinequality*******【Abstract】Thatinequalitiesinmathematicswasveryimportantroleandstatusandisevaluated,reasoning,mathematicalwayofthinkingisimportanttoinfiltrateintothesubjectismathcontentoftheimportantcomponentoftheinequalitiesintheprocessneedstobeusedinmanymathematicalthought,withmanyimportantmathematicalcontent。Inthispaper,Isummarizedsomemathematicalinequalityproofmethods.Inequalityinelementarymathematicalproofcommonlyuseincomparativelaw,forcommercial,analysis,synthesis,mathematicalinduction,thereduce-tiontoabsurdity,discriminant,function,Geometry,andsoon.Inequalityinhighermathematicsproofoftenusetheintermediatevaluetheorem,Taylorformula,theLagrangafunctionandsomefamousinequality,suchas:meaninequality,Kenseninequality,Johnsonin-equality,Helderinequality,andsoon.Inequalityproofmethodsgetmoreefficientandhelpusfurtherexploreandstudytheinequalityproof.Throughthestudyoftheseproofmethods,wecansolvesomepracticalproblems,developlogicalreasoningabilityanddemonstratedtheabilitytoabstractthinkingandgrowhardthinkingandgoodatthinkingofthegoodstudyhabit。【Keywords】inequalitycomparativelawmathematicalinductionfunction江西师范大学09届学士学位毕业论文3目录1引言............................................................................................................................42不等式证明的基本方法............................................................................................42.1比较法.............................................................................................................42.1.1作差比较法]1[...................................................................................42.1.2作商比较法..........................................................................................52.2分析法.............................................................................................................52.3综合法[2]..........................................................................................................62.4反证法]3[]4[.....................................................................................................62.5换元法..........................................................................................................82.5.1三角代换法..........................................................................................82.5.2增量换元法]5[......................................................................................92.6放缩法.............................................................................................................92.6.1“添舍”放缩......................................................................................92.6.2利用基本不等式]6[]7[........................................................................102.6.3分式放缩............................................................................................122.7迭合法...........................................................................................................132.8数学归纳法[8]................................................................................................132.9构造解析几何模型证明不等式...................................................................142.10判别式法[9]..................................................................................................142.11标准化法[10].................................................................................................152.12分解法.........................................................................................................153利用函数证明不等式..............................................................................................163.1利用函数单调性...........................................................................................163.2利用函数的极值...........................................................................................163.3利用函数的凹凸性.......................................................................................163.4利用中值定理...............................................................................................183.4.1利用拉格朗日中值定理]11[...............................................................183.4.2利用柯西中值定理]12[.......................................................................193．5利用泰勒公式.............................................................................................204小结..........................................................................................................................21参考文献：........................................................................................................