设函数f(x)=In(1+x)-2x/(x+2),证明:当x>0时,f(x)>0

问题描述:

设函数f(x)=In(1+x)-2x/(x+2),证明:当x>0时,f(x)>0

f(x) = ln(1+x) - 2x/(x+2)
f'(x) = 1/(1+x) - 2 [(x+2)-x]/(x+2)^2
= 1/(1+x) - 4/(x+2)^2
= [(x+2)^2 - 4(1+x)]/[(1+x)(x+2)^2]
= x^2/[(1+x)(x+2)^2] > 0( for x >0)
f(0) = 0
f(x)> f(0) = 0
ie f(x) > 0