利用插值法证明推广的柯西中值定理

The Proof of the Generalized Cauchy Mean-value Theorem with Method of Interpolation

借助插值的思想 ,首先给出函数 f( x)的泰勒公式的行列式表达式 ,推广了柯西中值定理 .据此拉格朗日中值定理、泰勒公式、罗必塔法则均是该结论的推论 ,从而对经典的中值定理、泰勒公式。

Having the aid of the insert value idea, we put out the expression of determinant with the Taylor polynomial of function f(x) . Then we gain the generalized Cauchy mean-value theorem. Thus, Lagrange mean-value theorem and Taylor formula and L′Hospital rule are its corollaries. Therefore, we unified prove the mean-value theorem and Taylor (formula) and L′Hospital rule.