HMMT 二月 2006 · CALC 赛 · 第 7 题
HMMT February 2006 — CALC Round — Problem 7
题目详情
- Find all positive real numbers c such that the graph of f : R → R given by f ( x ) = 3 x − cx has the property that the circle of curvature at any local extremum is centered at a point on the x -axis. ∫ π/ 3 2
解析
- Find all positive real numbers c such that the graph of f : R → R given by f ( x ) = 3 x − cx has the property that the circle of curvature at any local extremum is centered at a point on the x -axis. √ 3 Answer: 2 √ ′ 2 Solution: The equation 0 = f ( x ) = 3 x − c has two real roots: ± c/ 3. Let √ √ ′′ a := c/ 3. As f ( − a ) = − 6 c/ 3 < 0, f has a unique local maximum at x = − a. 2 Because f has half-turn symmetry about the origin, it suffices to consider this local extremum. The radius of curvature at any local extremum is 1 1 r ( x ) = = , ′′ | f ( x ) | 6 | x | so the condition in the problem is equivalent to r ( − a ) = f ( − a ) 1 2 = − a ( a − c ) 6 a 2 2 1 = 6 a ( c − a ) = 2 c (2 c/ 3) √ c = 3 / 2 .