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Hessian matrix in taylor expansion help

  1. Feb 6, 2012 #1
    1. The problem statement, all variables and given/known data

    Find the critical point(s) of this function and determine if the function has a maxi-
    mum/minimum/neither at the critical point(s) (semi colons start a new row in the matrix)

    f(x,y,z) = 1/2 [ x y z ] [3 1 0; 1 4 -1; 0 -1 2] [x;y;z]


    2. Relevant equations



    3. The attempt at a solution
    I'm fairly certain this is the second derivative of a taylor series expansion so 3rd term. So the matrix [3 1 0; 1 4 -1; 0 -1 2] is the Hessian. What I do not know now is how to get the maximum/minimum/neither or the critical points from the hessian.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 6, 2012 #2
    The critical points are such that all the partial derivatives are 0
     
  4. Feb 6, 2012 #3
    So does that mean where 3x+y=0, x+4y-z=0, and -y+2z=0?
     
  5. Feb 6, 2012 #4
    yes you are right
     
  6. Feb 6, 2012 #5
    Confirming that I have to solve it as a system of equations?
     
  7. Feb 6, 2012 #6

    Ray Vickson

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    Science Advisor
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    Yes, of course. That is exactly how critical points are found, in general.

    RGV
     
  8. Feb 7, 2012 #7
    I will get that all the values are equal to zero if I solve that?
     
  9. Feb 7, 2012 #8

    Ray Vickson

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    Science Advisor
    Homework Helper

    Try it and see.

    RGV
     
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