What is output noise or uncertainty (error propagation) for a min or a max function?

Question

For an input vector X = {${x_1 , x_2, ..., x_i, ..., x_n}$} with input noise as a variance-covariance matrix. If I perform $max$(X). What is the output uncertainty or noise.

Also, is the assumption that if $x_i$ is the $max$ of X, then is the output uncertainty/noise the same as noise at $x_i$ location. How does covariance in input noise affect this estimate?