微分中值定理逆命题的讨论

The Inverse Proposition of Differential Mean Value Theorem

对于常见的三个微分中值定理(罗尔中值定理,拉格朗日中值定理,柯西中值定理)的逆命题何时成立的问题进行了讨论。对于f′(x)仅有一个零点的情况得到了使罗尔中值定理逆命题成立的充要条件;对于一般情况,也得到了一个有价值的充要条件,利用辅助函数推广了关于罗尔中值定理逆命题的有关结果,得到了拉格朗日中值定理与柯西中值定理逆命题成立的条件。

In this paper,the inverse proposition of differential mean value theorem is disscussed.First,when derived function has only one zero point,the necessary and succficent condition is obtained which make the inverse proposition of Rolle mean value theorem true.Second,the inverse proposition of Rolle theorem is studied when derived function has several zero points.At last,making use of auxiliary function,the necessary and succficent condition is obtained which make the inverse proposition of Lagrange(Cauchy)mean value theorem true.