HMMT 二月 2007 · CALC 赛 · 第 4 题

英文解析

4. [ 4 ] Find the real number α such that the curve f ( x ) = e is tangent to the curve g ( x ) = αx .
   2 x 2 ′
   Answer: e / 4 . Suppose tangency occurs at x = x . Then e = αx and f ( x ) = 2 αx . On the 0
   0 0 0
   ′ 20
   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
   x 2 20
   x = 2. Then α = e / ( x ) = e / 4.0
   2 ′′ ′ ′ 200