# Critical region and power function for hypothesis test

If $X_{1},X_{2},\ldots,X_{n}$ are independent, where $X_{i}$ is $N(\theta_{i},1)$ , find the critical region and power function for the Most Powerful test of size $\alpha$ of
$H_{0}:\theta_{1}=\cdots=\theta_{n}=0$ against $H_{1}:\theta_{1}=\cdots=\theta_{r}=\frac{1}{2} , \theta_{r+1}=\cdots=\theta_{n}=-\frac{1}{2}$