1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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.

Share this great discussion with others via Reddit, Google+, Twitter, or Facebook