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摘要不等式是数学学习中的一个极其重要的部分。从小学一年级最简单的数学学习开始，学生们就已逐渐接触不等式。从不等式的解法开始到不等式的证明，再到柯西不等式、均值不等式等重要不等式的应用，不等式贯穿于整个数学学习过程中。在历年的中考以及高考试题中关于不等式的题型屡见不鲜。在中学数学中不等式的证明方法有综合法、数学归纳法、构造法、反证法、比较法、分析法、换元法、放缩法等初等方法。而在高等数学中不等式的证明方法有利用函数最大值最小值、凹凸性、泰勒公式、单调性、微分中值定理等来证明或者发现不等式。不等式本身是比较抽象的，而且它对逻辑性的要求也比较高，另外它的证明非常灵活，证明方法多样化，但却没有固定的模式，因此对数学的发散思维和敏锐度要求也很高。教师们在解决不等式问题时要注意方式方法，讲究一题多解，在此过程中，不仅无形地培养了学生的探索精神，而且启发学生去创新，激发创造性思维，从而提高了学生的思维水平。 46696
Abstract
Inequality plays an extremely important role in mathematics learning, and students in first grade usually start learning inequality although the math is quite simple at that level. Inequality learning is throughout the mathematics learning process from inequality solution to inequality demonstration, from the application of Cauchy inequality to the mean inequality and it is often tested in the high school examination as well as the College Entrance Examination. In middle school, there is comprehensive law, scaling law, construction law, reductio ad absurdum, analysis law, substitution method, comparative law, and other mathematical induction applied to prove inequality in math learning. Nevertheless, in higher mathematics, the methods of inequality proof are the concave and convex functions, monotonicity, minimum, maximum differential mean value theorem and Taylor formula. Inequality itself is considered comparatively abstract, thus it demands high logical thinking and since there are variable and flexible ways to prove it, no fixed modes or patterns can be used. Hence, the pergent thinking and sensitivity are highly required here. When we are teaching inequality, we have to pay much attention to the solution of a problem, and in this process, students shall gain the spirit of exploration, the ability of innovation and creative thinking as well. Thereby, the level of students’ thinking will be improved by the invisible effort of math teachers.
毕业论文关键词：初等数学； 高等数学； 凹凸性；微分中值定理； 单调性； 最值和极值；泰勒公式；
Keyword: Elementary Mathematics；advanced mathematics； bump nature；Differential Mean Value Theorem;monotonicity；most value and extremum;Taylor formula;
目 录
1 引言 4
2 初等数学中常见的不等式证明方法 4
2.1 综合法 4
2.2 分析法 6
2.3 比较法 7
2.4 构造法 8
2.4.1 利用函数的单调性 8
2.4.2 构造复数利用复数向量有关性质 8
2.5 反证法 9
2.6 数学归纳法 10
2.7 换元法 12
2.7.1增量换元 12
2.7.2三角换元 13
2.7.3和差换元 14
2.7.4向量换元