HMMT 二月 2004 · CALC 赛 · 第 4 题

HMMT February 2004 — CALC Round — Problem 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 .

解析

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 ′ n n ≥ 0. We will show by induction that f ( a ) = a and f ( a ) = ( − sin a ) for all n . The n n case n = 0 is clear. Then f ( a ) = cos f ( a ) = cos a = a , and n n − 1 ′ ′ n − 1 n f ( 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 − 8 2 2 ′ 2 4 8 6 4 2 cos a = 1 − a , so f ( a ) = (1 − a ) = a − 4 a + 6 a − 4 a + 1. 1