I'm struggling with showing that the third derivative is strongly increasing on the interval. I tried finding the fourth derivative and its roots, but it made me more confused.
I think the inequality has to do something with the error approximation, but I still don't see how to get the expressions $\frac{1}{3}x^3$ and $\frac{112}{81}x^3$.
Hint Write $f(x)=\frac{1}{2} \ln(1+x) -\frac{1}{2} \ln(1-x)$. This will help you calculate the third and fourth derivative faster.
Use the fact that $1 \pm x \geq 0$ on your interval.