Suppose $X_1, \cdots, X_n \sim \mathcal{N}(\mu, \Sigma)$, where $\mu \in \mathbb{R}^d$ and $\Sigma \in S_n^+$. The likelihood of the sample is given by: $$L(\mu, \Sigma) = \prod_{i=1}^n \frac{(2 \pi)^{-d/2}}{|\Sigma|^{1/2}} \exp\left\{-\frac{1}{2}(x_i - \mu)^T \Sigma^{-1} (x_i - \mu) \right\}.$$
I want to choose $\mu$ and $\Sigma$ which maximize $L$. I know that these are given by the sample mean and sample covariance, but I'm wondering if I can prove this using the KKT conditions? This is a constrained optimization problem since $\Sigma$ must be P.S.D. Do the KKT conditions apply here?