罗尔定理的推广

Extension of Rolle's Theorem

本文将罗尔定理“函数ｆ（ｘ）在闭区间［ａ，ｂ］连续，在开区间（ａ，ｂ）可导，且ｆ（ａ）＝ｆ（ｂ）；则至少存在一点ξ∈（ａ，ｂ），使ｆ′（ξ）＝０．”推广为：“函数ｆ（ｘ）在开区间（ａ，ｂ）可导，且ｌｉｍｘ→ａ＋ｆ（ｘ）＝ｌｉｍｘ→ａ－ｆ（ｘ）＝Ａ，其中ａ、ｂ、Ａ均可为有限数或无穷大，则至少存在一点ξ∈（ａ，ｂ），使ｆ′（ξ）＝０．”分别叙述为“推广１”、“推广２”、“推广３”，给出证明并用实例验证其正确性．

In this paper, the authors confirm that the conclusion of Rolle's Theorem is still correct if the original conditions are changed. The details are as follows: If a function f(x) is derivable in an open interval (a,b) and  lim x→a +f(x)= lim x→b -f(x)=A in which a,b and A are real numbers or infinitely great, then, there exists inside the open interval (a,b), at least one point ξ, so that the derivative f′(x) vanishes, that is f′(ξ)=0.At the same time, the authors define their theorems as 'Extension Ⅰ'. 'Extension Ⅱ' and 'Extension Ⅲ' and they also justify the correctness of their theorems by practical examples.