微积分在不等式证明中的应用

Applications of Calculus in the Proof of Inequalities

不等式是数学的重要内容,证明不等式的方法多种多样,有些不等式用初等方法来证明需要较高的技巧,甚至有时有些不等式根本无法用初等方法来证明.而有时利用高等数学中微积分的有关知识来证明不等式,可以使证明的思路变得简单,技巧性降低.在此总结出三个可直接用于证明不等式的命题,阐述如何利用高等数学中函数的单调性、拉格朗日中值定理、函数的板值与最值、函数凹凸性、泰勒公式、积分中值定理及其性质来证明不等式.

Inequality is an important part of mathematics.There are many ways proving inequality,and proving certain inequalities requires higher skills when using elementary methods.However,sometimes it is impossible to prove inequalities by elementary methods,and sometimes it is easy to prove inequalities by calculus of higher mathematics.This paper gives three proposition for proving inequalities and explains how to prove inequalities by the monotonicity of function,the Lagrange mean value theorem,the extreme value of function,the maximum and minimum value of function,concave and convex of function,Taylor formula,and the integral mean value theorem.