Results are out! Find what you need...fast. Get quick advice or join the chat
Hey! Sign in to get help with your study questionsNew here? Join for free to post

Local maxima, minima, and saddle points

Announcements Posted on
    • Thread Starter
    • 0 followers
    Offline

    ReputationRep:
    Show that if a>b>c>0 then the function:

     f(x,y,z) = (ax^2 +by^2 + cz^2 )\exp(-x^2-y^2-z^2)

    Can someone please help me with this question I have spent hours on it and have made no progress..
    • 5 followers
    Offline

    ReputationRep:
    Then the function what?
    • Thread Starter
    • 0 followers
    Offline

    ReputationRep:
    Show that if a>b>c>0 then the function:

     f(x,y,z) = (ax^2 +by^2 + cz^2 )\exp(-x^2-y^2-z^2)

    has two local maxima, one local minima and four saddle points.

    Can someone please help me with this question I have spent hours on it and have made no progress..
    • 5 followers
    Offline

    ReputationRep:
    I reckon there is more to it than the second derivative test and algebra. However, I can't think of the right idea yet.
    • 98 followers
    Offline

    ReputationRep:
    What working out have you done so far?
    • 4 followers
    Offline

    ReputationRep:
    (Original post by Neilding91)
    Show that if a>b>c>0 then the function:

     f(x,y,z) = (ax^2 +by^2 + cz^2 )\exp(-x^2-y^2-z^2)

    Can someone please help me with this question I have spent hours on it and have made no progress..
    1. write down the partal derivatives and take them equal zero.
    2. Solve them simoultaneously
    a) first solution is trivial (0,0,0)
    b)then arrange the first equation to x^2 and substitute into the others
    You will get y=0 and z=0 so x^2=1
    this gives 2 soluutions
    c) because of simmetry similary to above for y and z you will get more 4 solutions
    So you have 7 stationary points
    3. Determine the second partial derivatives
    4. Create the Hessian matrix for each given (x,y,z) value (7 matrices)
    5. For every matrix:
    Calculate the eigen values
    a) When the matrix positive definite (all eigen value is positive)
    then there minimum at that point where the matrix was calculated
    b) when the matrix negative definite there is maximum
    c) and when the matrix has both negative and positive eigen values or zero
    then there is a saddle at the given point
    • 5 followers
    Offline

    ReputationRep:
    (Original post by ztibor)
    ...
    That looks like my shopping list -- can't follow it either; too long.
    • 4 followers
    Offline

    ReputationRep:
    I haven't tried it yet but I cant see why brute force wouldn't work in trying to calculate the maxima and minima etc.

    EDIT: as ztbor did. Sounds like a reasonable method.
    • 4 followers
    Offline

    ReputationRep:
    (Original post by gff)
    That looks like my shopping list -- can't follow it either; too long.
    ... can you suggest what to follow
    According the theorem for existence of stationary point in multivariable functions:
    grad( f(\vec r))=\vec 0 (depending on that grad(f) is exists)
    To determine the type of this point you have to use the eigen values for Hessian matrix.

Reply

Submit reply

Register

Thanks for posting! You just need to create an account in order to submit the post
  1. this can't be left blank
    that username has been taken, please choose another Forgotten your password?
  2. this can't be left blank
    this email is already registered. Forgotten your password?
  3. this can't be left blank

    6 characters or longer with both numbers and letters is safer

  4. this can't be left empty
    your full birthday is required
  1. By joining you agree to our Ts and Cs, privacy policy and site rules

  2. Slide to join now Processing…

Updated: March 26, 2012
New on TSR

GCSE mocks revision

Talk study tips this weekend

Article updates
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.