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HMMT 二月 2007 · CALC 赛 · 第 2 题

HMMT February 2007 — CALC Round — Problem 2

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 3 ] Determine the real number a having the property that f ( a ) = a is a relative minimum of f ( x ) =
    4 3 2
    x − x − x + ax + 1.
解析

英文解析

  1. [ 3 ] Determine the real number a having the property that f ( a ) = a is a relative minimum of f ( x ) =
    4 3 2
    x − x − x + ax + 1.
    ′ 3 2
    Answer: 1 . Being a relative minimum, we have 0 = f ( a ) = 4 a − 3 a − 2 a + a = a (4 a + 1)( a − 1).
    Then a = 0 , 1 , − 1 / 4 are the only possibilities. However, it is easily seen that a = 1 is the only valuesatisfying f ( a ) = a .