HMMT 二月 2007 · CALC 赛 · 第 4 题
HMMT February 2007 — CALC Round — Problem 4
题目详情
- [ 4 ] Find the real number α such that the curve f ( x ) = e is tangent to the curve g ( x ) = αx . 2 ′′ ′ ′ 2
解析
- [ 4 ] Find the real number α such that the curve f ( x ) = e is tangent to the curve g ( x ) = αx . 2 x 2 ′ 0 Answer: e / 4 . Suppose tangency occurs at x = x . Then e = αx and f ( x ) = 2 αx . On the 0 0 0 0 ′ 2 other hand, f ( x ) = f ( x ), so αx = 2 αx . Clearly, α = 0 and x = 0 are impossible, so it must be that 0 0 0 x 2 2 0 x = 2. Then α = e / ( x ) = e / 4. 0 0 2 ′′ ′ ′ 2