设函数f(X)的定义域为R+,且有:1.f(1/2)=1,2.对任意正实数x,y都有f(X*y)=f(x)+f(Y),3.f(x)为减函数(1)求证:当x∈[1,正无穷)时,f(X)≤0(2)求证:当x,y属于R+,都有f(x/y)=f(X)-f(Y)(3)解不等式:f(-x)+f(3-x)≥-2
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![设函数f(X)的定义域为R+,且有:1.f(1/2)=1,2.对任意正实数x,y都有f(X*y)=f(x)+f(Y),3.f(x)为减函数(1)求证:当x∈[1,正无穷)时,f(X)≤0(2)求证:当x,y属于R+,都有f(x/y)=f(X)-f(Y)(3)解不等式:f(-x)+f(3-x)≥-2](/uploads/image/z/3619798-70-8.jpg?t=%E8%AE%BE%E5%87%BD%E6%95%B0f%28X%29%E7%9A%84%E5%AE%9A%E4%B9%89%E5%9F%9F%E4%B8%BAR%2B%2C%E4%B8%94%E6%9C%89%3A1.f%281%2F2%29%3D1%2C2.%E5%AF%B9%E4%BB%BB%E6%84%8F%E6%AD%A3%E5%AE%9E%E6%95%B0x%2Cy%E9%83%BD%E6%9C%89f%28X%2Ay%29%3Df%28x%29%2Bf%28Y%29%2C3.f%28x%29%E4%B8%BA%E5%87%8F%E5%87%BD%E6%95%B0%281%29%E6%B1%82%E8%AF%81%3A%E5%BD%93x%E2%88%88%5B1%2C%E6%AD%A3%E6%97%A0%E7%A9%B7%29%E6%97%B6%2Cf%28X%29%E2%89%A40%282%29%E6%B1%82%E8%AF%81%EF%BC%9A%E5%BD%93x%2Cy%E5%B1%9E%E4%BA%8ER%2B%2C%E9%83%BD%E6%9C%89f%EF%BC%88x%2Fy%29%3Df%28X%29-f%28Y%29%283%29%E8%A7%A3%E4%B8%8D%E7%AD%89%E5%BC%8F%EF%BC%9Af%EF%BC%88-x%EF%BC%89%2Bf%EF%BC%883-x%29%E2%89%A5-2)
设函数f(X)的定义域为R+,且有:1.f(1/2)=1,2.对任意正实数x,y都有f(X*y)=f(x)+f(Y),3.f(x)为减函数(1)求证:当x∈[1,正无穷)时,f(X)≤0(2)求证:当x,y属于R+,都有f(x/y)=f(X)-f(Y)(3)解不等式:f(-x)+f(3-x)≥-2
设函数f(X)的定义域为R+,且有:1.f(1/2)=1,2.对任意正实数x,y都有f(X*y)=f(x)+f(Y),3.f(x)为减函数
(1)求证:当x∈[1,正无穷)时,f(X)≤0
(2)求证:当x,y属于R+,都有f(x/y)=f(X)-f(Y)
(3)解不等式:f(-x)+f(3-x)≥-2
设函数f(X)的定义域为R+,且有:1.f(1/2)=1,2.对任意正实数x,y都有f(X*y)=f(x)+f(Y),3.f(x)为减函数(1)求证:当x∈[1,正无穷)时,f(X)≤0(2)求证:当x,y属于R+,都有f(x/y)=f(X)-f(Y)(3)解不等式:f(-x)+f(3-x)≥-2
(1)证明:由f(xy)=f(x)+f(y)取y=1得,f(x)=f(x)+f(1),所以f(1)=0,根据f(x)为减函数,所以x≥1时f(x)≤f(1)=0
(2)证明:当x≠0时,f(1)=f(x*1/x)=f(x)+f(1/x)=0,所以f(1/x)=-f(x).
那么当x,y属于R+时,f(x/y)=f(x)+f(1/y)=f(x)-f(y)
由(2)得f(1/2)=f(1)-f(2)=-f(2)=1,即f(2)=-1,所以f(4)=f(2*2)=f(2)+f(2)=-2
那么f(-x)+f(3-x)=f(x(x-3))≥f(4),
f(x)为减函数,所以x(x-3)≤4,=> -1≤x≤4
函数f(X)的定义域为R+,所以-x>0,3-x>0=>x