HMMT 二月 2005 · CALC 赛 · 第 8 题
HMMT February 2005 — CALC Round — Problem 8
题目详情
英文原题
- If f is a continuous real function such that f ( x − 1) + f ( x + 1) ≥ x + f ( x ) for all x ,
∫
2005
what is the minimum possible value of f ( x ) dx ? 1
解析
英文解析
- If f is a continuous real function such that f ( x − 1) + f ( x + 1) ≥ x + f ( x ) for all x ,
∫
2005
what is the minimum possible value of f ( x ) dx ?
Solution: 20100121
Let g ( x ) = f ( x ) − x . Theng ( x − 1) + x − 1 + g ( x + 1) + x + 1 ≥ x + g ( x ) + x,
or g ( x − 1) + g ( x + 1) ≥ g ( x ) . But now,
g ( x + 3) ≥ g ( x + 2) − g ( x + 1) ≥ − g ( x ) .
Therefore
∫ ∫ ∫
a +6 a +3 a +6
g ( x ) dx = g ( x ) dx + g ( x ) dxa a a +3
∫
a +3 = ( g ( x ) + g ( x + 3)) dx ≥ 0 .
It follows thata
∫ ∫
2005 6 n +7333
∑
g ( x ) = g ( x ) dx ≥ 0 ,
1 6 n +1
n =0
so that
[ ]
2005
∫ ∫ ∫
2 2
2005 2005 2005
x 2005 − 1
f ( x ) dx = ( g ( x ) + x ) dx ≥ x dx = = = 2010012 .
2 2
1 1 1
Equality holds for f ( x ) = x .1