HMMT 二月 2002 · CALC 赛 · 第 10 题

英文解析

10. A continuous real function f satisfies the identity f (2 x ) = 3 f ( x ) for all x . If
    ∫ ∫
    1 2
    f ( x ) dx = 1, what is f ( x ) dx ?
    0 1
    ∫ ∫
    2 1
    Solution: 5 Let S = f ( x ) dx . By setting u = 2 x , we see that f ( x ) dx =
    1 1 / 2
    ∫ ∫ ∫
    1 2 1 / 2
    f (2 x ) / 3 dx = f ( u ) / 6 du = S/ 6. Similarly, f ( x ) dx = S/ 36, and in general
    1 / 2 1 1 / 4
    ∫ ∫
    n − 1
    1 / 2 1
    f ( x ) dx = S/ 6 . Adding finitely many of these, we have f ( x ) dx = S/ 6 + S/ 36 +nn n
    1 / 2 1 / 2
    ∫
    n n 1
    · · · + S/ 6 = S · (1 − 1 / 6 ) / 5. Taking the limit as n → ∞ , we have f ( x ) dx = S/ 5. Thus
    S = 5, the answer.0 2