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Critical Points & their Nature of a Multivariable Function

  1. Oct 18, 2012 #1
    1. The problem statement, all variables and given/known data
    f(x,y) = xy(9x^2 + 3y^2 -16)

    Find the critical points of the function and their nature (local maximum, local minimum or saddle)


    2. Relevant equations



    3. The attempt at a solution
    I have partially differentiated the equation into:

    fx = 27yx^2* + 3y^3 -16y
    fy = 9x^3 + 9xy^2 - 16x

    How do I go on from there though?
     
  2. jcsd
  3. Oct 18, 2012 #2

    Zondrina

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    Hey there, welcome to the forum.

    So, recall that a critical point of a function f(x) occurs when the derivative f'(x) = 0.

    Now translating that to the multivariate case is not very different. Set fx = 0 and fy = 0 and then solve for your critical points.
     
  4. Oct 18, 2012 #3

    Ray Vickson

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    You solve the equations
    27*y*x^2 + 3*y^3 - 16*y = 0
    9*x^3 + 9*x*y^2 - 16*x = 0.
    The first one says either y = 0 or 27*x^2 + 3*y^2 - 16 = 0, and this last one is the equation of an ellipse in (x,y) space. The second one says either x = 0 or 9*x^2 + 9*y^2 -16 = 0. What kind of geometric figure does that describe in (x,y)-space?

    RGV
     
    Last edited: Oct 18, 2012
  5. Oct 18, 2012 #4
    Thanks alot for the welcome and the help. Greatly appreciated.
    Just a question:
    For both the equations:
    fx = 27*y*x^2 + 3*y^3 -16*y = 0
    and
    fy = 9*x^3 + 9*x*y^2 - 16*x = 0

    Am I supposed to solve both equations simultaneously to find the stationary points?

    I tried using a calculator to solve both equations simultaneously and I've gotten 9 stationary points.

    Am I on the right track?
     
  6. Oct 18, 2012 #5

    Zondrina

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    Ill give you a hint, use elimination to solve both. Solve for y in the first equation to get y=0 and y = ± something else.
     
  7. Oct 18, 2012 #6

    Ray Vickson

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    Answer to both questions is yes. The geometric representation I suggested before shows why there are 9 solution points.

    RGV
     
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