# Critical points in calculus

1. Oct 8, 2009

### 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?

2. Oct 8, 2009

### 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

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

3. Oct 9, 2009

### squenshl

Wow, 27 critical points.