Definitions first: $X=[1,2]$ $Y=[-1,1]$ and $f(x,y):=x+xy$ and the question I asked myself would be to calculate this: $\inf_Y\sup_Xf(x,y)$
As far as I have understood it you start by fixing $y$. What I would do is to have 3 cases:
1.)$y>0$
2.)$y=0$
3.)$y<0$
For the first case we would have $\inf_Y(2+2y)=0$
Second case: $\inf_Y2=2$
Now it is getting a bit more complicated if $y<0$ then the part $xy$ becomes smaller if $x$ is getting bigger or closer to $2$ So what do I now? Like thats the part Im having problems with, how do you take the sup now? How do you differentiate? I guess I am on the right track, maybe I am thinking too much and it is straightforward. I would love to see a good and long explanation going really int depth, such that I could use it for different examples, as well. Thanks in advance!