Question

定义在区间(0,正无穷)上的函数f(x)满足对任意实数x.y有f(x^y)=yf(x)

若a>b>c>1,且a,b,c成等差数列，求证f(a)f(c)<f(b)^2
若f(1/2)<0,求证f(x)为增函数

Answer 1

先证明　
若a>b>c>1,且a,b,c成等差数列，求证f(a)f(c)<f(b)^2

设公差为d, d>0, 则 a= b+d, c = b-d
再设 a=b^p, c = b^q,
由a>b>c>1知p,q都是正数, 且p!=q　
f(a)f(c) = f(b^p)f(b^q) = pq f(b)^2

现在证明 pq <1
p = logb.a = logb.(b+d)
q = logb.c = logb.(b-d)
√(pq) < (p+q)/2
= 1/2*( logb.(b+d)+ logb.(b-d))

= 1/2*( logb.(b+d)(b-d) )
= 1/2 *(logb.(b²-d²))
< 1/2 *(logb.(b²-0))
=1/2 *2*logb.b
=1
所以
f(a)f(c) = f(b^p)f(b^q) = pq f(b)^2 <1*f(b)^2

　

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再证f(x)为增函数

任意区间(0,正无穷)上的x,y, 若y>x
log1/2. y < log1/2. x
f(x) = f((1/2)^(log1/2. x)) = (log1/2. x)f(1/2)
f(y) = f((1/2)^(log1/2. y)) = (log1/2. y)f(1/2)

f(y)-f(x) = (log1/2. y -log1/2. x)f(1/2) >0
所以

f(x)为增函数　

追问

设 a=b^p, c = b^q,
啥意思

追答

把 a、c 记成 b的指数形式
这是为利用　f(x^y)=yf(x) 这个已知条件做准备