Maximize $ 2^{(-x)} + 2^{(-y)}$ subjected to certain conditions

Question

I am reading through convex optimization and I came across this following problem:

\begin{align*} \max \text{ } & 2^{-x}+2^{-y}\\ \text{s.t. } & \frac{1}{1+x}+\frac{1}{1+y}\leq b\\ & x\geq0\\ & y\geq0 \end{align*}

I have tried a few different approaches but none seem to work. How should I approach this problem?

Also, is there a way to solve the generalized version of this problem with the original constraints modified to include terms for the new variables (lets say $N$ variables.) ? (i.e. $x_i \geq 0$ and $\frac{1}{1 + x_1} + \frac{1}{1 + x_2} ... + \frac{1}{1 + x_N} \leq b $)

Answer 1

If $b\leq0$, it is easily verified that the program is infeasible. If $b\geq2$, it is easily verified that an optimal solution is $x=y=0$. Therefore, suppose $0<b<2$.

Edit: I removed my erroneous solution but am not deleting this answer since there are some useful comments below.