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Calculating critical points and classifying them

  1. Jul 19, 2010 #1
    1. The problem statement, all variables and given/known data

    Find all critical points and classify them


    2. Relevant equations

    f(x,y) = sin(x)sin(y)sin(x+y)

    0<=x,y<=Pi


    3. The attempt at a solution

    fx=sinysin(2x+y) and fy=sinxsin(2y+x)

    Therefore critical points are at:

    x=Pi/3(2n-m) , y=Pi/3(2m-n) where n>=1, m<=2, n,m belong to integer set (Z)

    fxx = sin(2x+2y)-sin(2x)
    fyy = sin(2x+2y)-sin(2y)
    fxy = sin(2x+2y)

    Now, to classify the critical points I was simply going to test for:

    fxxfyy-fxy^2

    If <0 it is a saddle point etc and substitute for x the definitions above to obtain the equation we can use to classify with

    Is this correct? Is there a better way? Would greatly appreciate comments. Thanks
     
  2. jcsd
  3. Jul 19, 2010 #2
    Yep, that's the second derivative test. So it's definitely correct (assuming you have that "etc" part right in your comment). As for a better way...there might be something intuitive about this function in particular that makes it special. I don't see anything, but I'm not sure. However, in general, the second derivative test is the way to go.
     
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