高考试题:已知函数f(x)=x2+2/x+alnx(x>0),f(x)的导函数是f’(x),对任意两个不相等的正数x1﹑x2,证明: (Ⅰ)当a≤0时,[f(x1)+f(x2)]/2>f[(x1+x2)/2];(Ⅱ)当a≤4时,|f’(x1)-f’(x2)|>|x1-x2|.该题可以运用不等式和导数的有关知识给出证明.在这里提出这样的问题:能否对题目中给出的a的条件作出进一步的加强,使得(Ⅰ)﹑(Ⅱ)仍然成立呢?为了探讨这个问题,首先给出一个定义和一个定理:定义(函数凸凹性):已知函数f(x)在区间(a,b)有定义, Entrance exam questions: known function f (x) = x2 +2 / x + alnx (x> 0), f (x) the derivative function is f ’(x), for any two non-equal positive x1 ﹑ x2 , Prove that: (1) when a≤0, [f (x1) + f (x2)] / 2> f [(x1 + x2) / 2] x1) -f ’(x2) |> | x1-x2 |. This question can be proved by applying the related knowledge of inequalities and derivatives. Here we ask whether the conditions of a given in the title can be further In order to explore this problem, we first give a definition and a theorem: Definition (Function convexity and concavity): The known function f (x) in the interval (a, b) Have definition,