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HMMT 十一月 2008 · GEN1 赛 · 第 1 题

HMMT November 2008 — GEN1 Round — Problem 1

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 2 ] Find the minimum of x − 2 x over all real numbers x .
    2009
解析

英文解析

  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.
    Alternate method: The graph of y = x − 2 x is a parabola that opens up. Therefore, the minimum 2 − ( − 2) − boccurs at its vertex, which is at = = 1. But 1 − 2 · 1 = − 1, so the minimum is − 1.2
    2 a 2
    2009