Power function of normal mean test

Let $X_1, X_2, \dots, X_n$ ($n>1$) be a random sample from a $N(\mu, 1)$ distribution, where $\mu\in\mathbb{R}$ is unknown. Let $0 < \alpha < 1$. To test the hypothesis $H_0: \mu=0$ against $H_1: \mu=\delta$, where $\delta > 0$ is a constant, let $\beta$ denote the power of the size $\alpha$ test that rejects $H_0$ if and only if $\frac{1}{n}\sum_{i=1}^n X_i > c_\alpha$, for some constant $c_\alpha$. Then which of the following statements is/are true?