Prove that for
Define: f(x)=ln(1+x) and g(x)=1/{1+x^2} and use Mean value theorem - Wikipedia, the free encyclopedia on where .
Not sure... trying it...
Edit:
It's works!
Here is one way to do it. Multiply by 1+x (which doesn't change the inequality since 1+x>0). Then we want to show that
Since f(0) = 0 it will be sufficient to show that f(x) is decreasing in that interval. So we want to show that its derivative is negative. But
We want to show that for . To do that, repeat the previous argument. Since f'(0)=0 it is sufficient to show that f'(x) is increasing in that interval. So we want to show that its derivative is positive. But
which is clearly positive for x>–1.