Functions of Several Variables — Question 2

Question 2 – Solution

Strategy The subtracted squares are nonnegative; solve $f=c$ for the level sets.

Range and extrema The maximum is $\boxed{9}$ at $(1,-2)$. There is no minimum because $f\to-\infty$ as $\|(x,y)\|\to\infty$, so the range is $\boxed{(-\infty,9]}$.

Level curves For $c<9$, $$4(x-1)^2+(y+2)^2=9-c,$$ an ellipse centered at $(1,-2)$ with semiaxes $\sqrt{9-c}/2$ and $\sqrt{9-c}$. For $c=9$ the level is one point; for $c>9$ it is empty.