HMMT 二月 2010 · CALC 赛 · 第 5 题

英文解析

5. [ 4 ] Let the functions f ( α, x ) and g ( α ) be defined as
   ( )
   ∣
   4 xα
   ∣
   d f
   ∣2
   f ( α, x ) = g ( α ) =
   ∣
   x − 1 dx 4
   x =2
   Then g ( α ) is a polynomial in α . Find the leading coefficient of g ( α ).
   ( )
   1 xα
   Answer: Write the first equation as ( x − 1) f = . For now, treat α as a constant. From this
   16 2
   equation, repeatedly applying derivative with respect to x gives
   ( ) ( )
   α − 1
   α x
   ′
   ( x − 1) f + f =
   2 2
   ( )
   ( ) ( )
   α − 2
   α α − 1 x
   ′′ ′
   ( x − 1) f + 2 f =
   2 2 2
   ( ) ( )
   ( ) ( )
   α − 3
   α α − 1 α − 2 x