Let $X_{1}, \ldots, X_{n}$ be independent samples from the exponential distribution with density $f(x ; \lambda)=\lambda e^{-\lambda x}$ for $x>0$, where $\lambda$ is an unknown parameter. Find the critical region of the most powerful test of size $\alpha$ for the hypotheses $H_{0}: \lambda=1$ versus $H_{1}: \lambda=2$. Determine whether or not this test is uniformly most powerful for testing $H_{0}^{\prime}: \lambda \leqslant 1$ versus $H_{1}^{\prime}: \lambda>1$.