HMMT 二月 2006 · 代数 · 第 10 题

英文解析

10. Determine the maximum value attained by
    4 2
    x − x
    6 3
    x + 2 x − 1
    over real numbers x > 1.
    Answer:1
    Solution: We have the following algebra:6
    4 21
    x −
    x − xx
    1=
    6 3
    x + 2 x − 13
    x + 2 −
    x 3
    x −1 =x
    ( ) ( )
    1 13
    x − + 2 + 3 x −
    x xx −1
    x 1
    ( ) ( )
    ≤ =
    1 1
    3 x − + 3 x −6
    x x
    ( ) ( )
    1 13
    where x − + 1 + 1 ≥ 3 x − in the denominator was deduced by the AM-GMx x
    √
    1 1+ 5
    inequality. As a quick check, equality holds where x − = 1 or when x = .
    x 2 4