How to find the lower bound of $f(x, y) = -2xy+x^2y^2+x^2$? $f(x, y) = -2xy+x^2y^2+x^2, x\in \mathbb{R}, y\in \mathbb{R}$, how to find its lower bound?
Here are my thoughts, I don't know if it is rigorous.
$f(x, y) = -2xy+x^2y^2+x^2=(xy-1)^2+x^2-1$, as $(xy-1)^2 \geq 0$, $x^2\geq0$, and they can not get to $0$ at the same time, therefore $f(x, y) > -1$.
Therefore, the lower bound of $f(x, y)$ is $-1$.

As you stated the function is $f(x, y)=(xy-1)^2+x^2-1$, and $x^2\ge0$, and $(xy-1)^2\ge0$. so it has a minimum when both of the inequalities are at their smallest
now $x^2$ is minimised when $x=0$, and at $x=0$, $(xy-1)^2=(-1)^2=1$, therefore $f(0,y)=0+1-1=0$,
But looking at $(x,y)=(\frac{1}{2},1)$, therefore $f(\frac{1}{2},1)=(\frac{1}{2}-1)^2+\frac{1}{2}^2-1=\frac{1}{4}+\frac{1}{4}-1=-\frac{1}{2}$ so minimising along $x=0$ won't give us the minima of the function
So maybe tying along the line where $(xy-1)^2$ is minimised (ie when $xy-1=0$) and seeing if it also allows you to minimises $x^2$
hopefully that helps
