 Solution

The Neyman-Pearson lemma compares two simple hypotheses with densities $f_0,f_1$. Among all tests of size at most $\alpha$, a most powerful test rejects when $f_1(x)/f_0(x)>c$, accepts when it is below $c$, and may randomize on equality to attain the prescribed size. The threshold and any boundary randomization are chosen so the rejection probability under $H_0$ is $\alpha$.

The key inequality is $({\varphi }^{\ast }-\varphi )(f_1-c{f}_0)\ge 0$ pointwise for the likelihood-ratio test ${\varphi }^{\ast }$. Integration shows its power is at least that of any competing test with no larger size. With continuous likelihood-ratio statistics, boundary randomization is unnecessary.