The level surface of the function $f(x,y,z) = (x^2+y^2)^{-1/2}$ are
a) Circles centered at the origin b) spheres centered at the origin c) cylinders around the z-axis d) upper halves of spheres centered at the origin
I had chosen a) asmy answer, but it turns out the answer is c). Could someone help me understand why? Thank you so much!
On
$f(x,y,z)= K \quad \Rightarrow \quad \frac{1}{\sqrt{x^2+y^2}} =K \quad \Rightarrow \quad x^2+y^2 = \frac{1}{\sqrt{K}}$
Therefore the level surfaces of $f$ have equation $x^2+y^2 = \frac{1}{\sqrt{K}}$, which is indeed a circle in $\mathbb{R}^2$ or a cylinder in $\mathbb{R}^3$. Since $f$ is defined on a subset of $\mathbb{R}^3$, it is necessarily a cylinder here.
If the function was $f(x,y) = \frac{1}{\sqrt{x^2+y^2}},$ your answer would be correct.
The answer is c because the function only depends on $x$ and $y$. You can move freely on the $z$ direction without changing the value of $f$.