HMMT 二月 2004 · CALC 赛 · 第 4 题

英文解析

4. Let f ( x ) = cos(cos(cos(cos(cos(cos(cos(cos x ))))))), and suppose that the number a
   ′
   satisfies the equation a = cos a . Express f ( a ) as a polynomial in a .
   8 6 4 2
   Solution: a − 4 a + 6 a − 4 a + 1
   This is an exercise using the chain rule. Define f ( x ) = x and f ( x ) = cos f ( x ) for
   0 n n − 1
   ′ nn ≥ 0. We will show by induction that f ( a ) = a and f ( a ) = ( − sin a ) for all n . Thecase n = 0 is clear. Then f ( a ) = cos f ( a ) = cos a = a , andnnn n − 1
   ′ ′ n − 1 nf ( a ) = f ( a ) · ( − sin f ( a )) = ( − sin a ) · ( − sin a ) = ( − sin a )
   n − 1
   n n − 1
   8 2
   ′ 8
   by induction. Now, f ( x ) = f ( x ), so f ( a ) = ( − sin a ) = sin a . But sin a = 1 −
   2 2 ′ 2 4 8 6 4 28
   cos a = 1 − a , so f ( a ) = (1 − a ) = a − 4 a + 6 a − 4 a + 1. 1