HMMT 二月 2011 · ALGCALC 赛 · 第 3 题

英文解析

3. Let f : R → R be a differentiable function such that f (0) = 0, f (1) = 1, and | f ( x ) | ≤ 2 for all real
   ∫
   numbers x . If a and b are real numbers such that the set of possible values of f ( x ) dx is the open 1
   interval ( a, b ), determine b − a .0
   Answer: Draw lines of slope ± 2 passing through (0 , 0) and (1 , 1). These form a parallelogram with 3
   vertices (0 , 0) , ( . 75 , 1 . 5) , (1 , 1) , ( . 25 , − . 5). By the mean value theorem, no point of ( x, f ( x )) lies outside 4
   this parallelogram, but we can construct functions arbitrarily close to the top or the bottom of theparallelogram while satisfying the condition of the problem. So ( b − a ) is the area of this parallelogram,
   which is .3 4