= Solution
The normalized density is
$$
f_\theta(x)=(\theta+1)x^\theta\mathbf1_{(0,1)}(x).
$$
For $\theta=1$ against $\theta=0$, the sample likelihood ratio is
$$
\frac{L(1)}{L(0)}
=2^n\prod_{i=1}^nX_i
=2^ne^{-S},
\qquad
S=-\sum_{i=1}^n\log X_i.
$$
It is strictly decreasing in $S$. Under $H_0$, parts (a) and (b) give
$$
S\sim\Gamma(n,1).
$$
If $q_\alpha$ is the lower $\alpha$-quantile of this gamma distribution, the <Neyman-Pearson lemma> gives the most powerful size-$\alpha$ critical region
$$
\boxed{S\leq q_\alpha}.
$$
For any fixed $\theta>0$,
$$
\frac{L(\theta)}{L(0)}
=(\theta+1)^ne^{-\theta S}
$$
is again strictly decreasing in the same statistic $S$. Thus the same critical region is most powerful against every $\theta>0$ and is consequently a <uniformly most powerful test> of $H_0:\theta=0$ against $H_1:\theta>0$.
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