微分中值定理的反问题

INVERSE PROBLEM OF DIFFERENTIAL MEAN VALUE THEOREM

本文证明了下述结果:定理对任意ξ∈(a,b),若1°f(x),g(x)在ξ点的某邻域上连续且在ξ点可微;2°F(x)=(f(x)-f(ξ))/(g(x)-g(ξ))在ξ点的某邻域内为x的严格递增函数(除ξ点外);3°g′(ξ)>0则在(a,b)内可找两点x_1 ,x_2:x_1<ξ<x_2,使得(f(x_2)-f(x_1))/(g(x_2)-g(x_1))=f′(ξ)/g′(ξ)

In this paper, we prove the results as follows,Theorem For any e(a,b), if 1. f(x), g(x) are continuous on a neighborhood for L and differentiable at L, 2. F(x) = [f(x) - f(L)]/[(g(x)-g(L)] is strict increase function on a neighborhood for L (except for L); 3. g'(L)> 0, then there exist x1, x2e (a,b), x1<L<x2,such as