= Solution
Interpret $\phi(x)$ as the conditional rejection probability. The <Type I error> is $\alpha=E_0\phi(X)$, the <Type II error> is $\beta=E_1[1-\phi(X)]$, the size is $\alpha$, and the power is $1-\beta$. Their weighted sum is
$$
c\alpha+\beta=1+\int\phi(x)[cp_0(x)-p_1(x)]dx.
$$
Pointwise minimization over $0\le\phi\le1$ forces rejection where $p_1>cp_0$ and acceptance where $p_1<cp_0$, with arbitrary randomization on equality. Conversely those choices minimize every integrand. The equivalence holds almost everywhere, since changing a test on a common null set cannot change its errors. With both weights strictly positive, any simultaneous improvement of the two errors, strict in at least one, would lower this weighted sum. \b[Every such positive-threshold <likelihood ratio> test is admissible for the two simple hypotheses.]
For the specified densities the <likelihood ratio> is
$$
R_\theta(x)=\begin{cases}0,&0<x<\theta,\\\dfrac{x-\theta}{x(1-\theta)^2},&\theta\le x\le1.\end{cases}
$$
It increases strictly on $(\theta,1)$ from zero to $1/(1-\theta)$. For $0<c<1/(1-\theta)$, the <likelihood-ratio test> rejects when $x>\theta/[1-c(1-\theta)^2]$. Larger thresholds accept almost surely. Threshold zero rejects everywhere above $\theta$ and permits arbitrary rejection below $\theta$, where both weighted choices tie; values at individual endpoints do not affect errors. These describe all nonnegative-threshold <likelihood ratio> tests, including their limiting accept-all and reject-all cases.
The upper-tail test $\phi_k$ has size $1-k^2$. If $\theta<k$, it is a positive-threshold <likelihood ratio> test and is admissible. If $\theta=k$, it rejects exactly the alternative's support, with power one; no test can retain power one with smaller size, so it remains admissible. If $\theta>k$, it still has power one but wastes rejection probability on $(k,\theta)$, where the alternative density vanishes. The test $\phi_\theta$ has the same power and smaller size, so
$$
\boxed{\phi_k\text{ is admissible against fixed }\theta\iff\theta\le k.}
$$
Nevertheless $\phi_k$ is a <uniformly most powerful test> at size $1-k^2$ against all $0<\theta<1$: for $\theta<k$, apply the <Neyman-Pearson lemma>; for $\theta\ge k$, its power is already the maximum one. It is also admissible against the composite alternative. Indeed choose any $0<\theta<k$. Strict monotonicity of its <likelihood ratio> makes $\phi_k$ the unique positive-threshold minimizer up to null sets. A test dominating it for the composite problem would either improve the weighted simple risk at this $\theta$, impossible, or tie that risk and coincide almost everywhere. Since $p_0>0$ throughout $(0,1)$, such coincidence also precludes improvement at any other $\theta$.
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