= Solution
Let $\Delta=\theta_1-\theta_0>0$. For one observation from the unit-variance normal location family,
$$
\log\frac{f_{\theta_1}(x)}{f_{\theta_0}(x)}
=\Delta\left(x-\frac{\theta_0+\theta_1}{2}\right).
$$
The sample log-<likelihood ratio> is therefore $nT_n$. By the <Neyman-Pearson lemma>, the level-$\alpha$ most powerful test rejects for a sufficiently large likelihood ratio, equivalently when $T_n>C_\alpha$, with $C_\alpha$ chosen to give null rejection probability $\alpha$.
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