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HMMT 二月 2006 · 代数 · 第 10 题

HMMT February 2006 — Algebra — Problem 10

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

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