HMMT 二月 2007 · CALC 赛 · 第 9 题

英文解析

9. [ 7 ] g is a twice differentiable function over the positive reals such that
   3 ′ 4 ′′
   g ( x ) + 2 x g ( x ) + x g ( x ) = 0 for all positive reals x. (1)
   lim xg ( x ) = 1 (2)
   x →∞
   Find the real number α > 1 such that g ( α ) = 1 / 2.2
   6 3 ′ 4 ′′
   Answer: . In the first equation, we can convert the expression 2 x g ( x )+ x g ( x ) into the derivativeπof a product, and in fact a second derivative, by writing y = 1 /x . Specifically,
   ( ) ( ) ( )
   1 1 1
   3 ′ 4 ′′ − 3 ′ − 4 ′′
   0 = g ( x ) + 2 x g ( x ) + x g ( x ) = g + 2 y g + y gy y y
   ( ) [ ( )]
   1 d 1 − 2 ′
   = g + − y gy d y y
   ( ) [ ( )]
   1 d 12 = g + gy d y y 2
   ( )
   Thus g = c cos( y ) + c sin( y ) or g ( x ) = c cos(1 /x ) + c sin(1 /x ). Now the second condition gives 1
   1 2 1 2
   sin(1 /x )y
   1 = lim c x + c · = c + lim c x
   1 2 2 1
   x →∞ x →∞
   1 /x
   It must be that c = 0 , c = 1. Now since 0 < 1 /α < 1, the value of α such that g ( α ) = sin(1 /α ) = 1 / 2
   1 2
   is given by 1 /α = π/ 6 and so α = 6 /π .