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Hessian matrix.

  1. Feb 18, 2008 #1
    Please,check my solution.
    Find critical points of the function [tex]f(x,y,z)=x^3+y^2+z^2+12xy+2z[/tex]
    and determine their types (degenerate or non-degenerate, Morse index for non-





    Critical points are at

    x=24 y=-144 z=-1

    x=0 y=0 z=-1


    for x=24

    [tex]det\left|\begin{array}{l[cr]}144&12&0\\12&2&0\\0&0&2\end{array}\right|=288[/tex] non-degenerate

    for x=0

    [tex]det\left|\begin{array}{l[cr]}0&12&0\\12&2&0\\0&0&2\end{array}\right|=-288[/tex] non-degenerate
  2. jcsd
  3. Feb 18, 2008 #2
    keep going... are they minimum, maximum.. saddle points??
  4. Feb 19, 2008 #3
    Morse index

    for (0,0,-1)

    [tex]det\left|\begin{array}{l[cr]}-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0[/tex]

    [tex](2-\lambda )(-\lambda(2-\lambda)-144)=0[/tex]


    for (24,-144,-1)

    [tex]det\left|\begin{array}{l[cr]}144-\lambda &12&0\\12&2-\lambda &0\\0&0&2-\lambda\end{array}}\right|=0[/tex]

    [tex](2-\lambda )((144-\lambda)(2-\lambda)-144)=0[/tex]


    Is it right?What we can say about maximum,minimum and saddle points?
  5. Feb 19, 2008 #4
    i didnt check you're calculus, but find what the sign of eigenvalues mean and you'll get you're answer.

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