关于Rolle定理的评注

Comments on and Explanation to Rolle's Theorem

Rolle定理是微分中值的基本定理 ,在微积分学中起着重要作用。关于 Rolle定理有着多种形式的推广。本文得到了如下有意义的结论。设函数 y=f( x)在区间 I的内部可导 ,在区间 I上连续 ,若在区间 I上存在三个点 x1 ,x2 ,x3 ,x1 <x2 <x3 ,使f( x2 )≥ max{ f( x1 ) ,f( x3 ) }或者 f( x2 )≤ min{ f( x1 ) ,f( x3 ) } ,则在区间 I内至少存在一点ξ,使 f (ξ) =0。此结论在证明关于“ξ等式”时十分方便 ,简洁。不论在理论上 。

Rolle's theorem is the fundamental theorem for differential median and plays an important role in the study of differential. Rolle's theorem has been promoted in different ways. This paper holds that, let function y=f(x) is able to differentiate at the interval I and is in continuum at the interval; if there exist three points x 1, x 2, and x 3 (x 1,<x2 ,<x 3) and let f(x 2) ≥max{ f(x 1), f(x 3)}, or f(x 2) ≤max{f(x 1), f(x 3)}, then there must exist a point ( at the interval I where f(ξ)=0. This conclusion proves very convenient and concise in proving ξ equation.