MLE vs. unbiased estimators of $\sigma^2$ in $N(\mu,\sigma^2)$

Let $x_1,x_2,\dots,x_n$ be a random sample from $N(\mu,\sigma^2)$; $-\infty<\mu<\infty, \sigma^2>0$, where both $\mu$ and $\sigma^2$ are unknown. Which of the following statements are correct?

I. $\hat\mu=\bar x$ and $\hat\sigma^2=\dfrac1n\sum_{i=1}^n(x_i-\bar x)^2$ maximize the likelihood function $L(\mu,\sigma^2;x)$.

II. $\hat\mu=\bar x$ and $S^2=\dfrac1{n-1}\sum_{i=1}^n(x_i-\bar x)^2$ maximize the likelihood function $L(\mu,\sigma^2;x)$.

III. $\hat\sigma^2=\dfrac1n\sum_{i=1}^n(x_i-\bar x)^2$ is an unbiased estimator of $\sigma^2$.

IV. $S^2=\dfrac1{n-1}\sum_{i=1}^n(x_i-\bar x)^2$ is an unbiased estimator of $\sigma^2$.