Power function of a one-sided normal LRT

Let $X_{1},X_{2},…,X_{n}$ be a random sample from $N(\theta,1)$, where $\theta\in(-\infty,\infty)$ is unknown. Consider the problem of testing $H_{0}:\theta\le 0$ against $H_{1}:\theta>0$. Let $\beta(\theta)$ denote the power function of the likelihood ratio test of size $\alpha$ ($0<\alpha<1$) for testing $H_{0}$ against $H_{1}$. Then, which of the following statements is/are TRUE?