 Solution

For known ${\sigma }^2>0$, maximizing the normal likelihood over $\mu$ is equivalent to minimizing ${\sum }_i(X_i-\mu {)}^2$. The identityshows that the unique minimum occurs at $\boxed{\hat{\mu }=\overline{X}}$. Equivalently, the log-likelihood has second derivative $-n/{\sigma }^2<0$.