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Math Help - critical points of a multivariable function

  1. #1
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    critical points of a multivariable function

    Find all the critical points of f(x,y)= x^3-6xy+3y^2-9x+7

    I get (3,0) and (-1,0) Did I do the question correctly? Please and thanks.
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  2. #2
    Super Member ILikeSerena's Avatar
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    Re: critical points of a multivariable function

    Quote Originally Posted by Dobby View Post
    Find all the critical points of f(x,y)= x^3-6xy+3y^2-9x+7

    I get (3,0) and (-1,0) Did I do the question correctly? Please and thanks.
    It doesn't look like it.
    How did you get your answer?
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  3. #3
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    Re: critical points of a multivariable function

    f_x=3x^2-6x-9
    f_y=-6x+6y

    now solving f_y=0, x=y => substitute that into f_x and solve and get x=3 and x=-1 then plug it back into f_y to get the y values and thus the two C.P.'s
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  4. #4
    Super Member ILikeSerena's Avatar
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    Re: critical points of a multivariable function

    Quote Originally Posted by Dobby View Post
    f_x=3x^2-6x-9
    f_y=-6x+6y

    now solving f_y=0, x=y => substitute that into f_x and solve and get x=3 and x=-1 then plug it back into f_y to get the y values
    Go on.
    What are the y values?
    Thanks from Dobby
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  5. #5
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    Re: critical points of a multivariable function

    with x=3, f_y=-6(3)+6(3)=0?

    same with x=-1

    or should I be plugging it into x=y to get (3,3) and (-1,-1)
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  6. #6
    Super Member ILikeSerena's Avatar
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    Re: critical points of a multivariable function

    Quote Originally Posted by Dobby View Post
    with x=3, f_y=-6(3)+6(3)=0?


    same with x=-1


    or should I be plugging it into x=y to get (3,3) and (-1,-1)
    Yep!
    That looks much better!

    Note that you just filled in y=3 to get to f_y=0.
    Last edited by ILikeSerena; April 23rd 2012 at 04:58 PM.
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  7. #7
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    Re: critical points of a multivariable function

    I don't get it can you please give me a little more help I've exerted my efforts maximally

    Edit: I see thanks for your help ! very silly of me
    Last edited by Dobby; April 23rd 2012 at 05:01 PM.
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  8. #8
    Super Member ILikeSerena's Avatar
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    Re: critical points of a multivariable function

    Quote Originally Posted by Dobby View Post
    I don't get it can you please give me a little more help I've exerted my efforts maximally
    From f_y=0, you deduced that y=x.
    Afterward you substituted that in f_x=0 to find x.
    To find the corresponding y you have to use the equation you found for y.

    From the fact that f_y=0, it is not right to conclude that y must be 0.
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  9. #9
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    Re: critical points of a multivariable function

    Absolutely you are correct! It is very clear to me now! Thank your very much !
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  10. #10
    Super Member ILikeSerena's Avatar
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    Re: critical points of a multivariable function

    You are aware of the thanks button?
    Thanks from Dobby
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  11. #11
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    Re: critical points of a multivariable function

    No, I was not but I just discovered it and will very well press. Also, please if I take this a step further,

    Am I correct in my calculations that (3,3) is a local minimum and (-1,-1) is neither seeing as for (-1,-1) => (f_xx)(f_yy)-[f_xy]^2=0
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  12. #12
    Super Member ILikeSerena's Avatar
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    Re: critical points of a multivariable function

    Quote Originally Posted by Dobby View Post
    No, I was not but I just discovered it and will very well press. Also, please if I take this a step further,

    Am I correct in my calculations that (3,3) is a local minimum and (-1,-1) is neither seeing as for (-1,-1) => (f_xx)(f_yy)-[f_xy]^2=0
    Aha! You are taking it to the next level!

    Not quite.
    Yes, (3,3) is a local minimum, but as yet (-1,-1) is inconclusive.
    See for instance: Second partial derivative test - Wikipedia, the free encyclopedia
    This means that (-1,-1) could still be either a minimum, a maximum, or a saddle point.
    Last edited by ILikeSerena; April 23rd 2012 at 05:32 PM.
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