I am stuck with the following problem. I can find the absolute minimum or maximum of a function with 2 variables or more, but I can't prove that absolute values exist.
Assume we have the following function:
$$f(x,y)=\frac{x^2-y^2}{\pi^{x^2-y^2}}$$
I need to show that the absolute maximum and minimum values exist
My idea: For a function to have absolute min/max value, the domain set should be compact: that is, closed and bounded and the function be continuous. But I am really stuck here. I dont know how to show that the domain set is compact. Could anyone please help me with this matter?
Best Answer
a note: the domain need not be compact for a function to have an absolute minimum or maximum. Take the trivial function $g(x)=0$. This function has an absolute minimum equal to its absolute maximum, which is 0.
As Andre Nicolas mentions, you can make this into a one variable problem. Let $r=x^2+y^2$. Then let $\tilde{f}(r)=\frac{r}{\pi^r}$. Try taking the derivative and see where this leads you.