<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] Let $\delta x = y-x$ and $x\ge y$. Then
$$4^x(1+4^{\delta x}) = 2^{x+1}(1+2^{\delta x})$$
then from factorization we can claim that:
$$2x = x+1 ~\text{and}~~ 4^{\delta x} = 2^{\delta x}$$
So, we get $x = y = 1$.
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