基于变限积分函数的Cauchy-schwards不等式的证明

A Proof of Cauchy-schwards Inequality Based on the Integral Functions with a Variable Upper Limit

提出一种以变上限积分函数为工具构造辅助函数证明Cauchy-schwards不等式的新方法.与高等数学常见的两种证明方法相比,该方法充分利用了变上限积分函数的导数之符号对其单调性的昭示作用,对于学生熟悉变上限积分函数的函数角色、构造辅助函数的思维训练以及综合利用导数和积分知识有一定的积极作用.

This paper expolres a new proof of the Cauchy-schwards inequality,which constructs an auxiliary function based on integral functions with variable upper limits. Compared with two common proof methods in the advanced mathematics,the skill takes on three properties. Firstly,it makes full use of the derivative of the auxiliary function,whose sign indicates the monotonicity. Secondly,it provides a chance for the students to be familiar with the role of the integral functions with variable range and the construcion of auxiliary function. Thirdly,to some extent,it helps synthesize the derivatives and integral knowledge.