<s>[INST] Functions of several variables If $f(x,y) = x^2 + xy + y^2 - 3x + 4y - 5$. I know the domain is $\mathbb R^2$. How to determine the image of f is my issue.
 [/INST] Have a look at my answer here: https://math.stackexchange.com/a/3619647/399263
You can always translate conics to cancel terms in $x,y$.
$f(x+a,y+b)=x^2+(-3+b+2a)x+xy+y^2+(2b+4+a)y+[\cdots]$
Solve $\begin{cases}2a+b-3=0\\2b+4+a=0\end{cases}\iff \begin{cases}a=\frac{10}3\\b=-\frac {11}3\end{cases}$
$$f(x+a,y+b)=\overbrace{(x^2+xy+y^2)}^{\ge 0}-\dfrac{52}3$$
In this case $x^2+xy+y^2\ge 0$ is always positive (discriminant $-3y^2<0$) with equality for $x=y=0$.
Thus $f$ has a minimum for $(x+a,y+b)=(0,0)\iff (x=-a,y=-b)$ of value $-\dfrac{52}3$.
Also $f$ is unbounded above since for instance $f(x,0)=x^2-3x-5$ is unbounded above.
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