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

squenshl
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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?
 
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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

<br /> f(x,y,z) = x^2(x^2 -1) + y^2(y^2-1) + z^2(z^2-1) <br />
 
Wow, 27 critical points.
 
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