HMMT 二月 2005 · CALC 赛 · 第 1 题

HMMT February 2005 — CALC Round — Problem 1

专题: Contest Math / 竞赛数学
难度: L3
来源: HMMT

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题目详情

英文原题

Let f ( x ) = x + ax + b , with a 6 = b , and suppose the tangent lines to the graph of f atx = a and x = b are parallel. Find f (1).
∫ ∫
∞ ∞
cos u sin u

解析

英文解析

Let f ( x ) = x + ax + b , with a 6 = b , and suppose the tangent lines to the graph of f atx = a and x = b are parallel. Find f (1).
Solution: 1
′ 2 2 2 2 2
Since f ( x ) = 3 x + a , we must have 3 a + a = 3 b + a . Then a = b , and since a 6 = b ,
a = − b . Thus f (1) = 1 + a + b = 1.
∫ ∫
∞ ∞
cos u sin u