Normal likelihood ($L(\mu ,{\sigma }^2)=(2\pi {\sigma }^2{)}^{-n/2}\exp [-{\sum }_i(x_i-\mu {)}^2/(2{\sigma }^2)]$)

The likelihood function of independent observations from the same normal distribution. For fixed positive variance ${\sigma }^2$, its unique maximizing mean is the sample mean: ${\sum }_i(x_i-\mu {)}^2={\sum }_i(x_i-\overline{x}{)}^2+n(\mu -\overline{x}{)}^2$. This algebraic identity proves global optimality without relying only on a stationary derivative.