Recall that for some $f(x, y)$ to be a joint density function, two things must be satisfied:
- $f(x, y)$ must be non-negative for all $x, y$.
- The area under $f(x, y)$ must be 1.
The first part is easily satisfied by noting that $K$ must be non-negative.
The second part can be satisfied by setting the integral of $f(x, y)$ to 1:
$\int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}f(x, y)\,\mathrm{d}x\mathrm{d}y = 1$
Which can be simplified to:
$\int\limits_{0}^{1}\int\limits_{0}^{1}Kxy\,\mathrm{d}x\mathrm{d}y = 1$
At this point it shouldn't be difficult to solve for $K$.
For the $\max$ and $\min$ questions, you could observe that
$\max(a, b) = \begin{cases}a\quad a\ge b,\\b\quad \mathrm{otherwise}.\end{cases}$ $\min(a, b) = \begin{cases}a\quad a\le b,\\b\quad \mathrm{otherwise}.\end{cases}$
which ought to help.