拉格朗日中值定理及其应用

Lagrange Mean Value Theorem and its Application

为了便于能更好的理解和应用拉格朗日中值定理。本文主要通过介绍拉格朗日中值定理的定义、性质及其在各种问题中的应用来为拉格朗日中值定理做出解释说明。我们知道,拉格朗日中值定理阐述了函数改变量f(b)-f(a)与导数f'(x)之间的联系,使我们能够利用导数来研究函数,函数的上升、下降,求函数的极值,函数的凹凸性和拐点等可以利用它来解释。罗尔定理中函数在区间上的改变量f(b)-f(a)=0,所以说它可作为拉格朗日中值定理的特例。本文中例举了遇到ξ,η∈(a,b),且ξ≠η满足某种关系式时,要证明此类型的命题,常用一次或几次的拉格朗日中值定理。可以看到,只要合适应用的拉格朗日中值定理,较复杂的关系式证明就会显得容易许多。

In order to better understand and apply Lagrange mean value theorem.This paper mainly introduces the definition,properties and application of Lagrange mean value theorem to explain Lagrange mean value theorem.As we know,Lagrange mean value theorem describes the relationship between function change f(b)-f(a) and derivative f’(x),the rise and fall of function,find the extreme value of function,concave convex property and inflection point of function.It can be used to explain.In Rolle’s theorem,the change of function in the interval f(b)-f(a)=0,so it can be used as a special case of Lagrange mean value theorem.In this paper,we use one or several times Lagrange mean value theorem to prove this type of proposition when we encounter ξ,η∈(a,b) and ξ≠η.We can see that as long as the Lagrange mean value theorem is applied properly,it will be much easier to prove the complex relationship.