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HMMT 二月 2005 · CALC 赛 · 第 18 题

HMMT February 2005 — CALC Round — Problem 18

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. What is f ( x ) f ( x ) dx ? 0 ′ 2 ′′
解析
  1. What is f ( x ) f ( x ) dx ? 0 Solution: 486/5 ∫ ∫ 1 f (1) 1 ′ 2 2 0 = f ( x ) f ( x ) dx = u du = ( f (1) − f (0) ), and 2 0 f (0) ∫ ∫ 1 f (1) 1 2 ′ 2 3 3 18 = f ( x ) f ( x ) dx = u du = ( f (1) − f (0) ) . 3 0 f (0) The first equation implies f (0) = ± f (1). The second equation shows that f (0) 6 = f (1), 3 3 3 and in fact 54 = f (1) − f (0) = 2 f (1) , so f (1) = 3 and f (0) = − 3. Then ∫ ∫ 1 f (1) 1 1 486 4 ′ 4 5 5 f ( x ) f ( x ) dx = u du = ( f (1) − f (0) ) = (243 + 243) = . 5 5 5 0 f (0) ′ 2 ′′