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HMMT 十一月 2008 · GEN1 赛 · 第 1 题

HMMT November 2008 — GEN1 Round — Problem 1

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 2 ] Find the minimum of x − 2 x over all real numbers x . 2009
解析
  1. [ 2 ] Find the minimum of x − 2 x over all real numbers x . 2 2 2 2 Answer: -1 Write x − 2 x = x − 2 x + 1 − 1 = ( x − 1) − 1. Since ( x − 1) ≥ 0, it is clear that the minimum is − 1. 2 Alternate method: The graph of y = x − 2 x is a parabola that opens up. Therefore, the minimum − ( − 2) − b 2 occurs at its vertex, which is at = = 1. But 1 − 2 · 1 = − 1, so the minimum is − 1. 2 a 2 2009