HMMT 二月 2005 · 代数 · 第 7 题

HMMT February 2005 — Algebra — Problem 7

Let x be a positive real number. Find the maximum possible value of √ 2 4 x + 2 − x + 4 . x

解析

Let x be a positive real number. Find the maximum possible value of √ 2 4 x + 2 − x + 4 . x 2 √ Solution: 2 2 − 2 Rationalizing the numerator, we get √ √ 2 2 2 2 4 4 4 x + 2 − x + 4 x + 2 + x + 4 ( x + 2) − ( x + 4) · √ = √ 2 4 2 4 x x + 2 + x + 4 x ( x + 2 + x + 4) 2 4 x √ = 2 4 x ( x + 2 + x + 4) 4 √ = 1 2 4 ( x + 2 + x + 4) x 4 √ = . 2 4 2 x + + x + 2 x x Since we wish to maximize this quantity, we wish to minimize the denominator. By √ √ 2 4 2 AM-GM, x + ≥ 2 2 and x + ≥ 4, so that the denominator is at least 2 2 + 2. 2 x x Therefore, √ 2 4 √ x + 2 − x + 4 4 √ ≤ = 2 2 − 2 , x 2 2 + 2 √ with equality when x = 2.