= Solution
A statistic is sufficient when the <conditional distribution> of the observation given it does not depend on the parameter. It is minimal sufficient when it is, up to <null sets>, a <measurable function> of every <sufficient statistic>. For a dominated family, the likelihood-ratio criterion identifies two observations precisely when their <likelihood ratio> is independent of the parameter.
The Rao–Blackwell theorem says that conditioning a square-integrable <unbiased estimator> on a <sufficient statistic> preserves unbiasedness and cannot increase <variance>. The Cramér–Rao bound for an <unbiased estimator> of scalar $\theta$ is $\operatorname{Var}_\theta\widehat\theta\ge1/I(\theta)$. It requires parameter-independent support, differentiability, interchange of differentiation and integration, and finite positive <Fisher information>. More generally the numerator is the square of the derivative of the estimand. The bound follows from $E_\theta[\widehat\theta\,\partial_\theta\log f]=1$ and Cauchy–Schwarz.
For the uniform sample, the joint <likelihood> is $\theta^{-n}\mathbf1_{\{0<x_i<\theta\ \forall i\}}$. Its dependence on the observations is through $T=\max_iX_i$, proving sufficiency. Two such <likelihoods> have parameter-independent ratio exactly when their maxima coincide, proving minimality. The density of $T$ is $nt^{n-1}/\theta^n$ on $(0,\theta)$. Unbiasedness of $h(T)$ says
$$
\int_0^\theta nh(t)t^{n-1}\,dt=\theta^{n+1}.
$$
Differentiating this absolutely <continuous> identity gives $h(t)=(n+1)t/n$ <almost everywhere>. Conversely that function is unbiased. Every finite-variance <unbiased estimator>, after Rao–Blackwellization, must therefore give this same function of $T$. Its <variance> cannot exceed that of any such estimator. Thus
$$
\boxed{\widehat\theta_{\rm MVU}=\frac{n+1}{n}\max_iX_i,\qquad\operatorname{Var}\widehat\theta_{\rm MVU}=\frac{\theta^2}{n(n+2)}.}
$$
The Cramér–Rao argument does not apply because the support depends on $\theta$; in particular differentiation of the support boundary invalidates the regular score identity.
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