Statistical Inference — sufficiency of the sample minimum

Let $x_1,x_2,…,x_n$ be a random sample from a population with probability density function $f(x,\theta)=e^{-(x-\theta)}$, $x>\theta$, $0$ otherwise. For parameter $\theta$, consider the following statements: 1. $\min(x_1,x_2,…,x_n)$ is a sufficient statistic of $\theta$. 2. $\bar x=\frac{1}{n}\sum_{i=1}^n x_i$ is a minimum variance bound estimator of $\theta$. Which of the statements given above is/are correct?