4. (5%) Consider the testing problem $H_0: \theta = \theta_0$ versus $H_1: \theta \neq \theta_0$ based on i.i.d. $X_1, X_2, \ldots, X_n$ from $U(0, \theta)$. Show that the uniformly most powerful (UMP) test with rejection region $X_{(n)} > \theta_0$ or $X_{(n)} \le \theta_0 \alpha^{1/n}$ is a likelihood ratio (LR) test with size $\alpha$.