<s>[INST] Proving global minimum by lower bound of 2-variable function $f(x,y)=x^4+2x^2y+y^2-4x^2-8x-8y$ I would like to prove that the following function $f :\mathbb{R}^2\to\mathbb{R}$ has a global minimum:

$f(x,y)=x^4+2x^2y+y^2-4x^2-8x-8y=(x^2+y)^2-4(x^2+2x+2y)$

$f$ has strict local minimum at
$f(1,3)=-20$
I think that what I need to show is that $-20$ is a lower bound of this function, and then conclude that's a global minimum, but I didn't manage to do so.
Please advise.
Thank you.
 [/INST] My favorite way,
$$f(x,y)+20=y^2+y(2x^2-8)+x^4-4x^2-8x+20$$
$$\begin{align}\Delta_{\text{half}}&=(x^2-4)^2-(x^4-4x^2-8x+20)\\
&=-4(x-1)^2≤0.\end{align}$$
This means, $f(x,y)+20≥0.$
Hence, for minimum of $f(x,y)+20$,  we need to take $x=1$ and $y=4-x^2$, which gives $f(x,y)+20=0.$
Finally, we deduce that
$$\min\left\{f(x,y)+20\right\}=0~ \\
\text {at}~ (x,y)=(1,3)$$
$$\min\left\{f(x,y)\right\}=-20~ \\ \text {at}~ (x,y)=(1,3)$$
where $f(x,y)=x^4+2x^2y+y^2-4x^2-8x-8y.$

Small Supplement:
Using the formula
$$ay^2+by+c=a(y-m)^2+n$$
where $m=-\dfrac{b}{2a}, n=-\dfrac{\Delta}{4a}$
$$\begin{align}y^2+y(2x^2-8)+x^4-4x^2-8x+20=(y+x^2-4)^2+4(x-1)^2≥0.\end{align}$$
</s>