HMMT 二月 2002 · CALC 赛 · 第 9 题

英文解析

9. Suppose f is a differentiable real function such that f ( x ) + f ( x ) ≤ 1 for all x , andxf (0) = 0. What is the largest possible value of f (1)? (Hint: consider the function e f ( x ).)
   x ′ x ′ x
   Solution: 1 − 1 /e Let g ( x ) = e f ( x ); then g ( x ) = e ( f ( x ) + f ( x )) ≤ e . Integrating
   ∫ ∫
   1 1
   ′ xfrom 0 to 1, we have g (1) − g (0) = g ( x ) dx ≤ e dx = e − 1. But g (1) − g (0) = e · f (1),
   0 0
   ′ xso we get f (1) ≤ ( e − 1) /e . This maximum is attained if we actually have g ( x ) = e
   ′ − xeverywhere; this entails the requirement f ( x ) + f ( x ) = 1, which is met by f ( x ) = 1 − e .