Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 4 / 27K / a / Solution

The part of the log-likelihood depending on $\theta$ isCompleting the square separates the sum into a term independent of $\theta$ and $n(\overline{X}-\theta {)}^T{\Sigma }^{-1}(\overline{X}-\theta )$. Positive definiteness makes its unique minimum occur at $\overline{X}$. Since linear combinations of independent multivariate normal vectors are normal,The Hessian matrix $-n{\Sigma }^{-1}$ is negative definite, confirming the global likelihood maximum.