HMMT 二月 2011 · CALCCOMB 赛 · 第 3 题

HMMT February 2011 — CALCCOMB Round — Problem 3

Let f : R → R be a differentiable function such that f (0) = 0, f (1) = 1, and | f ( x ) | ≤ 2 for all real ∫ 1 numbers x . If a and b are real numbers such that the set of possible values of f ( x ) dx is the open 0 interval ( a, b ), determine b − a .

解析

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