= Solution
The <likelihood ratio> for the two binomial probabilities is
$$
\frac{f_{3/4}(y)}{f_{1/4}(y)}=3^{2y-5},
$$
strictly increasing in $y$. By the <Neyman-Pearson lemma>, reject in the highest upper tail. Under the null,
$$
\mathbb P_{1/4}(Y\geq4)=\frac{5\cdot3+1}{4^5}=\frac{16}{1024}=\frac1{64}.
$$
Hence \b[reject exactly when $Y\geq4$], with no randomization required. Its <statistical power> at the stated alternative is
$$
\boxed{\mathbb P_{3/4}(Y\geq4)=\frac{5\cdot3^4+3^5}{4^5}=\frac{81}{128}}.
$$
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