Functions of Several Variables — Question 9

Question 9 – Solution

Strategy Enforce positivity of the logarithm’s argument and solve $F=c$ by exponentiating.

Step 1: Domain We need $$4-x^2-y^2-z^2>0,$$ so the domain is the open ball $\boxed{x^2+y^2+z^2<4}$.

Step 2: Range The logarithm’s argument takes every value in $(0,4]$. Therefore $F$ takes every value in $$\boxed{(-\infty,\ln 4]}.$$ The maximum $\ln 4$ occurs at the origin; the function tends to $-\infty$ near the boundary.

Step 3: Levels From $F=c$, $$4-r^2=e^c\Longrightarrow\boxed{r=\sqrt{4-e^c}},\qquad c\le\ln 4.$$ These are concentric spheres, degenerating to the origin at $c=\ln 4$. As $c$ increases, $e^c$ increases and the radius decreases.