In my homework I need to find the density function of $X_{\max}$ and $X_{\min}$ where
$$X_{\min} = \min\{X_1(w), ..., X_n(w)\},$$
$$X_{\max} = \max\{X_1(w), ..., X_n(w)\}$$
and where the $X_i$, $i = 1,\ldots,n$, are i.i.d. random variables with a given density function f and distribution function F which are not relevant here.
There is a hint:
"How do you express $[X_{\min} > t]$ under usage of $[X_1 > t], ..., [X_n > t]$ for absolut steady distributions?"
How do I go about this? I have absolutely no clue.
I think you can use the following facts: $$ \mathbb{P} (X_\min \le x) = 1- \mathbb{P} (X_\min > x) = 1-\mathbb{P}(X_1 > x, \ldots, X_n > x). $$
For $X_\max$ you can write directly $$ \mathbb{P} (X_\max \le x) = \mathbb{P}(X_1 \le x, \ldots, X_n \le x). $$
Recall that, if the $X_i$'s are independent, you can further develop the last term (in both the two equations).