对“均值不等式的八种证法”再思考

Rethink on “Eight Proofs for Average Inequality”

均值不等式是高中数学的重要内容,是不等式的补充.本文对均值不等式的算术归纳法、局部调整法、排序原理、不等式法、几何方法、变量替换法、归纳原理、逐次调整法等八种证明方法进行了推广.选择算术——几何均值不等式作为研究对象,借助数学归纳法、伯努利不等式法、泰勒公式法等十二种方法对均值不等式进行了证明.

Average inequality is an important part of high school mathematics,which is a supplement to the inequality. In this paper,eight kinds of proof methods of mean inequality are extended,such as Mathematical induction,partial adjustment method,sorting principle,inequality method,geometric methods,variable substitution,the principle of induction,successive adjustment method,etc. The arithmetic-geometric mean inequality is taken as the object of study,the mean inequality is proved by twelve methods,such as Mathematical induction method,Bernoulli inequality method,and Taylor formula method.