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Critical points in calculus

  1. Oct 8, 2009 #1
    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 & [tex]\pm1/\sqrt{2}[/tex], y = 0 & [tex]\pm1/\sqrt{2}[/tex], z = 0 & [tex]\pm1/\sqrt{2}[/tex].
    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. jcsd
  3. Oct 8, 2009 #2
    You have the trivial all 0 and all [itex]\pm1[/itex] solutions, and imagine
    [tex]\begin{align*} x^2 &= 1-\epsilon,\\ y^2 &= 1 + \epsilon,\\ \epsilon&\in[0,1]\end{align*}[/tex]

    Then what happens to

    [itex]
    f(x,y,z) = x^2(x^2 -1) + y^2(y^2-1) + z^2(z^2-1)
    [/itex]
     
  4. Oct 9, 2009 #3
    Wow, 27 critical points.
     
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