How many critical points are there in a function with multiple variables?

[squenshl]

I was given f(x,y,z) = x⁴ + y⁴ + z⁴ - x² - y² - z².
I found that (or least I think it's these) x = 0 & $$\pm1/\sqrt{2}$$, y = 0 & $$\pm1/\sqrt{2}$$, z = 0 & $$\pm1/\sqrt{2}$$.
What I'm stuck with is exactly how much critical points are there, by the looks of things there are a few but I'm not too sure, how do I arrange them?

 

[trambolin]

You have the trivial all 0 and all $\pm1$ solutions, and imagine
$$\begin{align*} x^2 &= 1-\epsilon,\\ y^2 &= 1 + \epsilon,\\ \epsilon&\in[0,1]\end{align*}$$

Then what happens to

$f(x,y,z) = x^2(x^2 -1) + y^2(y^2-1) + z^2(z^2-1)$

 

[squenshl]

Wow, 27 critical points.