摘 要

無論在初等數(shù)學(xué)還是高等數(shù)學(xué)中,不等式都是十分重要的內(nèi)容.而不等式的證明則是不等式知識的重要組成部分.在本文中，我總結(jié)了一些數(shù)學(xué)中證明不等式的方法.在初等數(shù)學(xué)不等式的證明中經(jīng)常用到的有比較法、作商法、分析法、綜合法、數(shù)學(xué)歸納法、反證法、放縮法、換元法、判別式法、函數(shù)法、幾何法等等.在高等數(shù)學(xué)不等式的證明中經(jīng)常利用中值定理、泰勒公式、拉格朗日函數(shù)、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、赫爾德不等式等等.從而使不等式的證明方法更加的完善，有利于我們進一步的探討和研究不等式的證明. 通過學(xué)習(xí)這些證明方法，可以幫助我們解決一些實際問題，培養(yǎng)邏輯推理論證能力和抽象思維的能力以及養(yǎng)成勤于思考、善于思考的良好學(xué)習(xí)習(xí)慣.

關(guān)鍵詞：不等式；比較法；數(shù)學(xué)歸納法；函數(shù)等等











Abstract

No matter in elementary maths or higher in mathematics, inequality is very important content. The inequality proof is an important part of the inequality knowledge. In this paper, I summarized some mathematical proof of inequality technique. In elementary mathematics inequality for the evidence is often used as a comparison, the commercial law, analysis and synthesis, mathematical induction, reduction, zooming method, in yuan method, discriminant method, function method, geometric method, etc. In the higher mathematics inequality for often use the evidence of the mean value theorem, Taylor formula, Lagrange function, and some famous inequality, such as: mean, inequality cauchy inequality, Jason, inequality holder inequation, etc. So that inequality proof of the method is more perfect, be helpful for our further research and study of the inequality proof. By studying the identification method, can help us solve some practical problems, cultivate logical reasoning ability and the abstract thinking ability, and develop thinking, good at thinking of the good study habits.
Keywords: inequality; Comparison method; Mathematical induction; Function and so on .











目錄

摘要……………………………………………………………………1
Abstract………………………………………………………………2
0引言 …………………………………………………………………4
1利用函數(shù)證明不等式
1.1函數(shù)極值法…………………………………………………5
1.2單調(diào)函數(shù)法…………………………………………………5
1.3中值定理法…………………………………………………6
1.4利用拉格朗日函數(shù)法………………………………………6
2利用著名不等式
2.1利用均值不等式……………………………………………8
2.2利用柯西不等式……………………………………………9
2.3利用赫爾德不等式…………………………………………9
2.4利用詹森不等式……………………………………………10
3利用積分不等式的性質(zhì)
3.1積分不等式的性質(zhì)…………………………………………11
3.2積分不等式的證明…………………………………………12
參考文獻………………………………………………………………20
致謝……………………………………………………………………21