Suppose I have a C∞ function, which I wish to prove attains its maximum/minimum. First I must prove that the function is bounded at all. If I determine R, the region (of the plane in this case) where the function is strictly positive, and integrate over R to find a finite answer, can I say the function is bounded from above over R? I am sure this is true for R having finite measure, but am not sure exactly what the condition is for an infinite region, which leads to an improper Riemann integral. Once I am sure the function is bounded, I can just take some closed set about the sup and min, and use continuity to prove the function attains the extrema. The function in question is (x2 - y2)e(x2 + y2) but that is probably not important.