I have this problem which I'm having trouble solving. I have to find the minimum and maximum of the following set:
$$A=\left\{\frac{x-y}{x+y+3} \mid x\in[-1, 1], y\in[-1, 1]\right\}.$$
Now, I feel like this problem would be much easier if the variables from the numerator and denominator were different. I would just find the minimum and maximum of the numerator and denominator, and the maximum over the minimum would be the max of the set, and min over max would be the min of the set.
However, since the variables used are the same, I don't know how to go about solving this problem, and would appreciate some help.
Thank you!
we denoting by $$f(x,y)=\frac{x-y}{x+y+3}$$ the function defined on $$-1\le x\le1$$ and $$-1\le y\le 1$$ the we get the partial derivatives as $$\frac{\partial f(x,y)}{\partial x}=\frac{x+y+3-(x-y)}{(x+y+3)^2}$$ and $$\frac{\partial f(x,y)}{\partial y}=\frac{-(x+y+3)-(x-y)}{(x+y+3)^2}$$ Can you finish?