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HMMT 二月 2004 · CALC 赛 · 第 2 题

HMMT February 2004 — CALC Round — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. Suppose the function f ( x ) − f (2 x ) has derivative 5 at x = 1 and derivative 7 at x = 2.
    Find the derivative of f ( x ) − f (4 x ) at x = 1.
    √ √
    3 3
    3 2 3 2
解析

英文解析

  1. Suppose the function f ( x ) − f (2 x ) has derivative 5 at x = 1 and derivative 7 at x = 2.
    Find the derivative of f ( x ) − f (4 x ) at x = 1.
    Solution: 19
    Let g ( x ) = f ( x ) − f (2 x ). Then we want the derivative off ( x ) − f (4 x ) = ( f ( x ) − f (2 x )) + ( f (2 x ) − f (4 x )) = g ( x ) + g (2 x )
    ′ ′
    at x = 1. This is g ( x ) + 2 g (2 x ) at x = 1, or 5 + 2 · 7 = 19.
    √ √
    3 3
    3 2 3 2