HMMT 二月 2004 · CALC 赛 · 第 2 题
HMMT February 2004 — CALC Round — Problem 2
题目详情
- 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
- 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 of f ( 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