摘要作者主要介绍利用几何思想证明不等式的一些基本方法。不等式是高中数学的重要内容之一，中学不等式的问题主要有两大类，一类是含未知数的不等式求解问题；另一类是不等式证明问题。所谓证明不等式，意在推出这个不等式对其中所有允许值的未知数都成立或推出数值不等式成立。由于不等式在中学数学中占有重要地位，应此在历年高考中颇受重视。不等式的证明方法多样，因题而异，灵活多变，技巧性强，是一个具有很强挑战性的数学内容。在不等式的证明中，认真观察和分析不等式及条件，通过构造、联想和猜想，与相关知识结构融合，对不等式进行合理的几何含义表征，从而找到以形助数的方法来加以证明。47562

毕业论文关键词：不等式； 几何不等式； 构造法； 数形结合

Abstract Using geometric ideas to prove the inequalities is introduced by the author. Inequalities is one of the important contents of high school mathematics. The inequality problems are mainly two categories, one category is to solve problems containing variables; another is to prove the inequality. The so-called proof of inequalities intend to launch this inequality on which the variables all allowed values are established or launch a numerical inequality. Inequality plays an important role in mathematics teaching in high school. It was emphasized in the college entrance examination. The methods of Inequality proof is persified by the different problems and is flexible, skillful. So it’s a very strong challenges of mathematics contents. In the proof of the inequalities, careful observation and analysis the known conditions are necessary, therefore, construction, associative and conjecture, integration with the learned knowledge and giving out the reasonable geometric characterization of the inequalities, which will be found to help shape method to verify the inequality.

Keyword:  Inequality；  Geometric inequality；  Construction method；   combination of number and shape

目    录

1、绪论 5

   1.1不等式的发展历史 5

   1.2不等式研究的意义 5

2、不等式及其证明 6

   2.1不等式在中学数学中的地位 5

   2.2不等式的常用证明方法 6

       2.2.1比较法（作差法） 6

       2.2.2作商法 7

       2.2.3分析法（逆推法） 7

       2.2.4综合法 7

       2.2.5反证法 8

       2.2.6跌合法 8

       2.2.7放缩法 8

       2.2.8数学归纳法 9

       2.2.9换元法 9

       2.2.10三角迭代法 9

       2.2.11判别式法 10

3、构造法 10

   3.1构造函数模型 11

   3.2构造方程模型 11

   3.3构造向量模型