微分中值定理的推广

THE EXTENSION OF THE MEAN VALUE THEOREM FOR DIFFERENTIAL

本文将把微分中值定理推广到函数系及其高阶导数的情形,从而使中值定理具有更一般的形式.

In this paper,We extend the Mean Value Theorem to system of functions and its higher-order derivatives.That is thefollowing theorem:Theorem If f_i(x)(i=1,2,…,n;n≥3)are continuous on[a,b],f_i^(n-2)(x)(i=1,2,…,n)are existences in(a,b)Then there are certain real numbers (?)_1,(?)_2,…,(?)_((?)2),such thata<(?)_1<(?)_2<…<(?)_(n-2)<bandf_1(a) f_2(a) … f_n(a)f_1(b) f_2(b) … f_n(b)f'_1((?)_1) f'_2((?)_1) … f'_n((?)_1)=0.f'_1((?)_2) f'_2((?)_2) … f'_n((?)_2)… … … …f_1^(n-2)((?)_(n-2)) f_n^(n-2)((?)_(n-2)) … f_n^(n-2)((?)_(n-2))Notice that:1.If n=3,f_3(x)≡1,and f'_2(x)≠0x∈(a,b),Then we obtain the cauchy mean value theorem.2.If n=3,f_2(x)≡x,and f_3(x)≡1,x∈[a,b],Then we obtain the Lagrangian mean value theorem.Finally,We extend this theorem to infinite interval.