Likelihood ratio test for the rate of a shifted exponential

Let $X$ be a random sample of size one from the probability density function

$$f(x\mid\theta)=\begin{cases}\theta e^{-\theta(x-1)},&\text{if }x>1,\\0,&\text{otherwise,}\end{cases}$$

where $\theta(>0)$ is the unknown parameter. Suppose we want to test the null hypothesis $H_0:\theta=1$ against the alternative hypothesis $H_1:\theta\neq1$, based on the observed value $x$ of $X$. Then, which of the following statements are true?