Global minimum for a function on an unrestricted domain

Question

We have a function $f(x,y) = (\dfrac{3}{2} - x + xy)^2 + (\dfrac{9}{4} - x + xy^2)^2$ I know that $\nabla f(0,1) = \nabla f(3,0.5) = 0$ and that $f(3, 0.5) < f(0,1)$. I am wondering whether or not this is sufficient to decide that the global minimum is $(3, 0.5)$? Or do we need to check other conditions too?

Answer 1

Yes, it is sufficient (provided that there are no more critical points) by the following argument. Since the function is sum of squares of polynomials it follows that $f(x,y)\to +\infty$ as $x^2+y^2\to +\infty$. It makes it possible to conclude that the minimum, if exists, must attain within some bounded (compact) domain. In this domain, the minimum exists for sure from Weierstrass theorem. Hence the minimum in the original problem exists as well and is the same. When the existence is cleared, one can apply the necessary condition that the minimum is attained at a critical point. If you find all critical points, the minimum is one of them.