不等式证明.doc

学号: 09003040 不等式证明的若干方法学院名称巢湖学院专业名称: 数学与应用数学年级班别: 09数学姓名: 项俊指导教师: 陈佩树 20 11年03月本科毕业论文 1 不等式证明的若干方法摘要无论在初等数学还是高等数学中, 不等式都是十分重要的内容. ,、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,,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、;比较法;数学归纳法;函数 AL ot ofM ethods about I nequality P roof Abstract In elementary mathematics and higher mathematics, i nequalities are very important elem ents . Inequality is an ponent in the inequality proof. In this paper, I summa rized some mathematical inequality proof method s. Inequality in elementary mathematical proo monly use parative law, mercial, analysis, synthesis, mathematical induction, the reduce - tion to absurdity, discriminant, function, Geometry, and soon. Inequa lity in higher mathematics proof often use the intermediate value theorem, Taylor formula, the Lagranga func tion and some famous i nequality, such as: mean inequality, Ke ns en inequality , Johnson in- equality, Helder inequality, and nequality proof method s get more efficient and help us further explore and study the inequality proof . Through the study of these pro of methods, we can solve some practical problems, develop logical reasoning ability and demonstrated the ability to abstract thinking and grow hard thinking and good at thinking of the good study habit. Key word si nequality; c omparative l aw; m athematical induction; functio n 2 1常用方法 比较法(作差法) [1] 在比较两个实数 a 和b 的大小时,可借助 ba?:作差——变形——判断(正号、负号、零) . 变形时常用的方法有:配方、通分、因式分解、和差化积、应用已知定理、: 0?a ,0?b ,求证: ab ba??2 . 证明02 )(2 22 2????????baab baab ba , 故得ab ba??2 . 作商法在证题时,一般在 a ,b 均为正数时,借助 1?b a 或1?b a 来判断其大小,步骤一般为: 作商——变形——判断(大于 1或小于 1). 例2设0??ba ,求证: abbababa?. 证明因为0??ba , 所以1?b a ,0??ba . 而1?????????baab bab aba ba , 故 abbababa?. 分析法(逆推法) 从要证明的结论出发,一步一步地推导,最后达到命题的已知条件(可明显成立的不等