Finding critical points of a triple variable function

Question

What are the critical points of the function: $f(x,y,z)=\frac{1}{z^2+x^2+1+y^2}$? Identify as a local minima, maxima, or saddle points.

So I know you have to take the gradient and set it equal to $(0,0,0)$. That will get you all your critical points. How do I identify it as a local minima, maxima, or a saddle point?

Answer 1

One way is to find the Hessian and determine its curvature by looking at its eigenvalues.

If the eigenvalues are all positive, then the function as positive curvature at that point, and you've found a minima.

If the eigenvalues are all negative, then negative curvature. And if they're mixed, then it's a saddle point.

Answer 2

There are no minima and no saddle points, the only maxima is at $x=y=z=0$, with the value of $1$.

The reason:

The function is spherically symmetric to the origin, so we can substitute $r=\sqrt{x^2+y^2+z^2}$. Doing that, we can see, that we have essentially a function of $\frac{1}{r^2+1}$, where r is the distance of the point from the origin. This function is monotonically decreasing from $1$ to $0$, while $r$ is going from zero to infinite. Its derivative is never zero (except the origin).