Find critical points using spherical coordinate

Question

again!

Let $E:=\left\{ (x,y,z) \in \mathbb{R}^3 \mid x^2+y^2+z^2 \le 4 \right\}$ and $f:E \to \mathbb{R}$ the function define by $$f(x,y,z):= \frac{z}{1+x^2+y^2+z^2}.$$ How can I determine the critical points of this function?

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I suppose I can convert this function by using the spherical coordinate, $$f(x,y,z)=f(\rho, \varphi, \theta)=\frac{\rho\cos{\varphi}}{1+\rho^2}$$

Now, I don't know what to do. I even don't know if my convertion is alright.

Thanks for your help.

Answer 1

Your conversion is right. Now, find the critical points, evaluate its gradient in therms of $\rho, \phi, \theta.$