关于波利亚的中值定理问题

The Polya's Lagrange Mean Value Theorem

波利亚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 。

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) can one finds 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 conwexity point,one can get a positive answer to the above question,whichis an improvement of previous work.The problem for bivariables function is also discussed. [WT5FZ]