Consider i.i.d. observations $X_{1}, \ldots, X_{n} \sim N\left(\mu, sigma^{2} ight) $ with $\mu$ unknown and $\sigma^{2}$ known. For the problem $H_{0}: \mu=0$ vs $H_{i}: \mu>0$, the testing procedure that rejects $H_{0}$ whenever $\frac{ \sqrt{n} \bar{X}}{\sigma}>z_{1-\alpha}$ has the power function $P (\mu)=\mathbb{P}\left(N(0,1)+\frac{ \sqrt{n} \mu} {\sigma)>z_{1-\alpha} ight) $. Show $P\mu) $ is an increasing function by taking derivative. SP.PB. 114