 Solution

For every particular alternative mean ${\mu }_1>{\mu }_0$, the likelihood ratio is increasing in the same statistic $\overline{x}$. Its size-$\alpha$ threshold is fixed solely by the null distribution, so it is always the same $c_{\alpha }$ found in part (c), independent of ${\mu }_1$.

The single rejection region is therefore most powerful against every member of the composite alternative. Hence the test is uniformly most powerful for $H_1:\mu >{\mu }_0$.