Find a two-variable polynomial that is bounded below and does not ever achieve its infimum

Question

I have this function: $f(x,y)=(1−xy)^2+x^2$ which is bounded below by $0$. I know it does not have a global minimum. How do I check if it will ever achieve an infimum?

Answer 1

First observe that $f \neq 0$, because if $f=0$, then $xy=1$ and $x=0$ (which is not possible).

Now we claim that $f$ cannot achieve it's infimum (actually we will show that $\inf f=0$.)

Let $\epsilon >0$, then let $y=\frac{1}{\sqrt{\epsilon}}$ and $x=\sqrt{\epsilon}$. Thus $f\left(\sqrt{\epsilon},\frac{1}{\sqrt{\epsilon}}\right)=\epsilon$. So $f$ can achieve any value as close to $0$ as possible, but as stated above $f \neq 0$. So $0$ is the infimum of $f$ which it cannot achieve.

Answer 2

$f(\frac 1 n, n)=\frac 1 {n^{2}}$ so the infimum is $0$.