HMMT 二月 2010 · CALC 赛 · 第 5 题

HMMT February 2010 — CALC Round — Problem 5

[ 4 ] Let the functions f ( α, x ) and g ( α ) be defined as ( ) ∣ α x 4 ∣ d f 2 ∣ f ( α, x ) = g ( α ) = ∣ 4 x − 1 dx x =2 Then g ( α ) is a polynomial in α . Find the leading coefficient of g ( α ). 3 2

解析

[ 4 ] Let the functions f ( α, x ) and g ( α ) be defined as ( ) ∣ α x 4 ∣ d f 2 ∣ f ( α, x ) = g ( α ) = ∣ 4 x − 1 dx x =2 Then g ( α ) is a polynomial in α . Find the leading coefficient of g ( α ). ( ) α 1 x 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