<s>[INST] Find value range of $2^x+2^y$ Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$

I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I can only find one solution pair of $4^x+4^y = 2^{x+1} + 2^{y+1}$. It seems very far from solve this question... 
 [/INST] You have
$$(2^x+2^y)^2=4^x+4^y+2^{x+y+1}=2^{x+1}+2^{y+1}+2^{x+y+1}$$
If $a=2^x$, $b=2^y$, then
$$(a+b)^2=2(a+b+ab)>2(a+b)$$
so $a+b>2$.
I'm looking for an upper bound. If I find it, I'll edit this answer.
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