I was reading a paper and came across a section that claimed that if $y \in \mathbb{N}$, and if $x \in [0,1]$, then for the expression:
$$ \left(\frac{1}{2}+ \frac{x}{4}\right)^y $$
there exists a $x_0 \in [0,1]$ such that:
$$ \left(\frac{1}{2}+ \frac{x_0}{4}\right)^y = \left(\frac{1}{2}\right)^{y_1}\left(\frac{x_0}{4}\right)^{y_2} $$
where $y = y_1 + y_2$.
I am not able to see how this is true. Is there such a way to prove this? thanks.