关于G.POLYA的中值定理问题

The Polya's Lagrange Mean Value Theorem

G.Polya曾提出并否定回答了与 L agrange中值定理有关的问题 :对于 y=f( x) ,x∈ ( a,b)是否对任意的 ξ∈ ( a,b)都存在 x1,x2 ∈ ( a,b) ,x1<ξ<x2 ,使 f′( ξ) =f ( b) -f ( a)b-a ?作进一步讨论并证明了当 ξ为凹凸性点时 ,上述 x1,x2 存在 ,改进了已有的结论 ,同时还讨论了二元函数的相关问题。

G.Polya posed a question related with Lagrange's mean value theorem.For y=f(x) with x∈(a,b) and any ξ∈(a,b) one can find x_1,x_2∈(a,b),x_1<ξ<x_2,such that f′(ξ)=f(b)-f(a)b-a. It is proved that if ξ is a convexity point,one can get positive answer to the above question,which is an improvement of previous work.The problem for two variables functions is also discussed.