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UMP test is not unique here. Since the pdf $f$ has monotone likelihood ratio (MLR) in $X_{(n)}$, by Karlin-Rubin theorem a UMP size $\alpha$ test for testing $H_0:\theta=\theta_0$ against $H_1:\theta<\theta_0$ is $$\phi_0(x_1,\ldots,x_n)=\begin{cases}1&,\text{ if }x_{(n)}<\theta_0\alpha^{1/n} \\ 0&,\text{ otherwise }\end{cases}$$ Now ...


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