HMMT 二月 2006 · CALC 赛 · 第 7 题

英文解析

7. Find all positive real numbers c such that the graph of f : R → R given by f ( x ) =
   x − cx has the property that the circle of curvature at any local extremum is centered 3
   at a point on the x -axis.
   √
   Answer:3
   √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.
   Because f has half-turn symmetry about the origin, it suffices to consider this local 2
   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 tor ( − a ) = f ( − a ) = − a ( a − c )21
   6 a
   2 2
   1 = 6 a ( c − a ) = 2 c (2 c/ 3)
   √
   c = 3 / 2 .