Consider the function $2^{y}$. This is monotonically increasing, so to maximize $2^{y}$, you simply want to maximize $y$. Here, $y=x(1-x)=-x^{2}+x$ is a parabola opening downwards. The vertex of the parabola occurs at $x=(-1) /(-2)=1 / 2$, so the maximum value of the function is $2^{(1 / 2)(1 / 2)}=\sqrt[4]{2}$.