设f(x)在[0,1]上为非减函数,且满足下列三个条件f(0)=0;f(x/5)=0.5f(x);f(x)=2-f(1-x),则f(1/2012)=A0.5B0.25C0.0625D1/2012

Question

设f(x)在[0,1]上为非减函数,且满足下列三个条件f(0)=0;f(x/5)=0.5f(x);f(x)=2-f(1-x),则f(1/2012)=
A 0.5 B 0.25 C 0.0625 D 1/2012

Answer 1

f(0)=0;f(x/5)=0.5f(x);f(x)=2-f(1-x),
f(1)=2-f(0)=2,
f(1/2)=2-f(1/2)==> f(1/2)=1
f(1/5)=0.5*f(1)=1
f(1/25)=0.5*f(1/5)=1/2
f(1/125)=1/4
f(1/625)=1/8
f(1/3025)=1/16
f(1/2)=1
f(1/10)=1/2
f(1/50)=1/4
f(1/250)=1/8
f(1/1250)=1/16
∵f(x)在[0,1]上为非减函数
∴f(1/1250)≥f(1/2012)≥f(1/3025)
∴1/16≥f(1/2012)≥1/16
∴f(1/2012)=1/16=0.0625
选C