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Critical Points

  1. Aug 21, 2010 #1
    1. The problem statement, all variables and given/known data

    Determine the critical points f(x,y) = x^3y + xy

    2. Relevant equations



    3. The attempt at a solution

    fx(x,y) = 3x^2 + y
    3x^2y+y=0
    y(3x^2 +1) = 0
    y = 0
    3x^2=1
    x=1/3


    fy(x,y) = x^3 + x
    x^3 + x = 0
    x(x^2+1) = 0
    x=0
    x^2= -1
    x= -1

    I don't know how to take it from here as I have only seen this done where you substitute y values in equations for x. What are the critical points of the above? Thanks for the help!
     
  2. jcsd
  3. Aug 21, 2010 #2

    lanedance

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    Homework Helper

    so all viable combinations of the solutions you found above, where fx = fy = 0, are critical points

    if you want to exmaine the nature of the critical point you will need to consider a type of 2nd derivative test or the hessian matrix
     
  4. Aug 22, 2010 #3

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    So either y= 0 or x= 1/3.


    In particular, x= 1/3 does not satisfy this. In order that both equations be satisfied, you must have x= 0, y= 0 or x= -1, y= 0.

     
  5. Aug 23, 2010 #4

    ehild

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    Gold Member

    I suppose that you assumed 3x^2+1=0. From here, everything is wrong what you did. From 3x^2+1=0 does not follow that 3x^2=1.
    From 3x^2=1 does not come that x=1/3.

    X^2=-1 does not mean that x=-1. Do you know what x^2 means? Can be the square of any real number equal to a negative number?


    ehild
     
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