定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,n>0),且当x>1时 ,f(x)>0求:(1) f(1)的值(2) 求证:f(m/n)=f(m)-f(n)(3) 求证:f(x)在(0,+∞)上是增函数(4)
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![定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,n>0),且当x>1时 ,f(x)>0求:(1) f(1)的值(2) 求证:f(m/n)=f(m)-f(n)(3) 求证:f(x)在(0,+∞)上是增函数(4)](/uploads/image/z/10210721-41-1.jpg?t=%E5%AE%9A%E4%B9%89%E5%9C%A8%280%2C%E2%88%9E%29%E4%B8%8A%E7%9A%84%E5%87%BD%E6%95%B0f%28x%29%E6%BB%A1%E8%B6%B3f%28mn%29%3Df%28m%29%2Bf%28n%29%2C%28m%EF%BC%9E0%2C%E5%AE%9A%E4%B9%89%E5%9C%A8%280%2C%E2%88%9E%29%E4%B8%8A%E7%9A%84%E5%87%BD%E6%95%B0f%28x%29%E6%BB%A1%E8%B6%B3f%28mn%29%3Df%28m%29%2Bf%28n%29%2C%28m%EF%BC%9E0%2Cn%EF%BC%9E0%29%2C%E4%B8%94%E5%BD%93x%EF%BC%9E1%E6%97%B6+%2Cf%28x%29%EF%BC%9E0%E6%B1%82%EF%BC%9A%281%29+f%281%29%E7%9A%84%E5%80%BC%282%29+%E6%B1%82%E8%AF%81%EF%BC%9Af%28m%2Fn%29%3Df%28m%29-f%28n%29%283%29+%E6%B1%82%E8%AF%81%EF%BC%9Af%28x%29%E5%9C%A8%280%2C%2B%E2%88%9E%29%E4%B8%8A%E6%98%AF%E5%A2%9E%E5%87%BD%E6%95%B0%284%29)
定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,n>0),且当x>1时 ,f(x)>0求:(1) f(1)的值(2) 求证:f(m/n)=f(m)-f(n)(3) 求证:f(x)在(0,+∞)上是增函数(4)
定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,
定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,n>0),且当x>1时 ,f(x)>0
求:(1) f(1)的值
(2) 求证:f(m/n)=f(m)-f(n)
(3) 求证:f(x)在(0,+∞)上是增函数
(4) 若f(2)=1 解不等式f(x+2)-f(2x)>2
定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,定义在(0,∞)上的函数f(x)满足f(mn)=f(m)+f(n),(m>0,n>0),且当x>1时 ,f(x)>0求:(1) f(1)的值(2) 求证:f(m/n)=f(m)-f(n)(3) 求证:f(x)在(0,+∞)上是增函数(4)
(1)令m=2,n=1
f(2)=f(2)+f(1),所以f(1)=0
(2)令f(m/n)+f(n)=f[(m/n)*n]=f(m),所以f(m/n)=f(m)-f(n)
(3)令x2>x1>0
f(x2)-f(x1)=f(x2/x1)
因为x2/x1>0,x>1时f(x)>0,所以f(x2)-f(x1)>0
f(x)在(0,+ ∞)上单调递增
(4)f(x+2)>f(2x)+2=f(2x)
f(x+2)>f(2x)+2f(2),f(x+2)>f(2x)+f(4)
f(x+2)>f(8x)
所以x+2>0,2x>0
且x+2>8x,所以0