G1.2 Given that $(x-1)^{2}+y^{2}=4$, where $x$ and $y$ are real numbers. If the maximum value of $2 x+y^{2}$ is $b$, find the value of $b$. (Reference: $\mathbf{2 0 0 9}$ HI5, 2011 HI2)
$\begin{aligned} 2 x+y^{2} & =2 x+4-(x-1)^{2} \\ & =-x^{2}+2 x-1+2 x+4 \\ & =-x^{2}+4 x+3 \\ & =-\left(x^{2}-4 x+4\right)+7 \\ & =-(x-2)^{2}+7 \leq 7=b\end{aligned}$