Statistical Inference — checking claims about a fixed-width uniform location model

Let $X_1, X_2, X_3, ..., X_n$ be a random sample from $U(\theta,\theta+1)$. Which of the following statements is/are correct? 1. The estimator $\left(\bar X-\frac12\right)$ is the best unbiased estimator of $\theta$. 2. The joint probability density function of $X_1,...,X_n$ can be written as $f(x,\theta)=1$ if $\max_i x_i-1<\theta<\min_i x_i$; $0$ otherwise. Select the correct answer using the code given below: