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HMMT 十一月 2014 · 团队赛 · 第 2 题

HMMT November 2014 — Team Round — Problem 2

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [ 5 ] Let f ( x ) = x + 6 x + 7. Determine the smallest possible value of f ( f ( f ( f ( x )))) over all real numbers x .
解析
  1. [ 5 ] Let f ( x ) = x + 6 x + 7. Determine the smallest possible value of f ( f ( f ( f ( x )))) over all real numbers x . 2 2 Answer: 23 Consider that f ( x ) = x + 6 x + 7 = ( x + 3) − 2. So f ( x ) ≥ − 2 for real numbers x . Also, f is increasing on the interval [ − 3 , ∞ ) . Therefore f ( f ( x )) ≥ f ( − 2) = − 1 , f ( f ( f ( x ))) ≥ f ( − 1) = 2 , and f ( f ( f ( f ( x )))) ≥ f (2) = 23 . Thus, the minimum value of f ( f ( f ( f ( x )))) is 23 and equality is obtained when x = − 3 .