Find the maximum of $\frac{1}{1+|x|}+\frac{1}{1+|x-a|}$ Let $a>0$. Show that the maximum value of the function 
$$f(x)= \frac{1}{1+|x|}+\frac{1}{1+|x-a|}$$
is $$\frac{2+a}{1+a}.$$
really need some help with this thing 

Study what happens in each of the three regions $\{x<0\}$, $\{0\le x\le a\}$ and $\{x>a\}$.
