函数f(x)的定义域为R,并满足以下条件：1,对任意x属于R,有f(x)大于零

题目

函数f(x)的定义域为R,并满足以下条件：1,对任意x属于R,有f(x)大于零
2,对任意x,y属于R,有f(xy)=[f(x)]^y;
3,f(1/3)>1
问;1,求f(0)的值
2,求证：f(x)在R上是单调增函数
3,若a>b>c>0,且b^2=ac,求证：f(a)+f(c)>2f(b)

答案

1.令x=0 f(0)=[f(0)]^0=1 f(0)=12.取y>0 x=1/3 f(y*1/3)=f(1/3)^y 任取x1>x2 f(x1)/f(x2)=f(3x1/3)/f(3x2/3)=[f(1/3)]^3x1/[f(1/3)]^3x2=[f(1/3)]^3(x1-x2) 3(x1-x2)>0 f(1/3)>1[f(1/3)]^3(x1-x2)>1 f(x1)>f(x2) 所...