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 

$f(a) = \dfrac{2+a}{1+a}$, and $f(x) = f(a-x)$, it suffices to consider $f$ on $[0,\frac{a}{2}]$, and on this interval $f(x) = \dfrac{1}{1+x} + \dfrac{1}{1+a-x}$, and $f'(x) = -\dfrac{1}{(1+x)^2} + \dfrac{1}{(1+a-x)^2} \leq 0$, thus $f(x) \leq f(0) = \dfrac{2+a}{1+a}$.
