已知函数f(x)的定义域为[0,1]且同时满足:①对任意x∈[0,1]总有f(x)≥2;②f(1)=3;③若x1≥0,x

2024-12-18 02:25:52
推荐回答(1个)
回答1:

(Ⅰ)令x1=x2=0,
由③知f(0)=2f(0)-2?f(0)=2;
(Ⅱ)任取x1x2∈[0,1],且x1<x2
则0<x2-x1≤1,∴f(x2-x1)≥2
∴f(x2)-f(x1)=f[(x2-x1)+x1]-f(x1
=f(x2-x1)+f(x1)-2-f(x1)=f(x2-x1)-2≥0
∴f(x2)≥f(x1),则f(x)≤f(1)=3.
∴f(x)的最大值为3;
(Ⅲ)由Sn=?

1
2
(an?3)知,
n=1时,a1=1;当n≥2时,an=?
1
2
an+
1
2
an?1

an
1
3
an?1(n≥2),又a1=1,∴an
1
3n?1

f(an)=f(
1
3n?1
)=f(
1
3n
+
1
3n
+
1
3n
)=f(
2
3n
)+f(
1
3