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Homework Help: Critical points of function in 3 variables

  1. Aug 1, 2010 #1

    K29

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    1. The problem statement, all variables and given/known data
    I have the scalar function
    [tex]T(r)= 2x^{2} - 4x +y^2 +4y-3z^{2}[/tex] (r is obviously a vector)
    I have the critical point (1,-2,0) but I'm not sure how to work out if its a min/max/saddle. I'm familiar with doing it for functions in 2 variables.
     
    Last edited: Aug 1, 2010
  2. jcsd
  3. Aug 1, 2010 #2
    No, for functions of 3+ variables, you have to look at the eigenvalues of the Hessian matrix at the critical points.

    If all eigenvalues are positive, then crit. pt. is a minimum. If all negative, then crit. pt. is a maximum. If some of the eigenvalues are negative while the rest are positive, then the crit. pt. is a saddle point. If any of the eigenvalues are equal to zero, then our test is inconclusive.
     
  4. Aug 1, 2010 #3
    Oh, and the Hessian matrix is the nxn matrix of 2nd partial deriv's of your function, where n is the number of variables (3 in your case). Look it up on wiki if you're unfamiliar.
     
  5. Aug 1, 2010 #4

    K29

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    Ok thanks. I edited the question while you were replying, but I understand. To anyone else this will look a mess.
     
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