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Math Help - Finding global max and min of a function

  1. #1
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    Finding global max and min of a function

    I need to find the global maximum and minimum of the function
    (3x^2+2xy+y^2)/(x^2-2xy+3y^2).
    I did the partial derivatives, set them equal to zero, took the numerator since the function will only equal zero when the numerator is zero, and got the following:
    y(-x^2+2xy+y^2) = 0 and 0 = x(x^2-2xy-y^2)

    I know that (x,y) can't equal (0,0) because the function is undefined there, so I need to find other (x,y) so that -x^2+2xy+y^2 = 0 and x^2-2xy-y^2 = 0. I recognized that those are both quadratic equations, so I used the quadratic formula to find y in terms of x and got
    y = (-1(+/-)2^.5)x

    I don't know what to do from there. How do I find the critical points now that I have that equation for y?

    Thanks!
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  2. #2
    Newbie
    Joined
    Nov 2011
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    Re: Finding global max and min of a function

    Quote Originally Posted by tubetess123 View Post
    I need to find the global maximum and minimum of the function
    (3x^2+2xy+y^2)/(x^2-2xy+3y^2).
    I did the partial derivatives, set them equal to zero, took the numerator since the function will only equal zero when the numerator is zero, and got the following:
    y(-x^2+2xy+y^2) = 0 and 0 = x(x^2-2xy-y^2)

    I know that (x,y) can't equal (0,0) because the function is undefined there, so I need to find other (x,y) so that -x^2+2xy+y^2 = 0 and x^2-2xy-y^2 = 0. I recognized that those are both quadratic equations, so I used the quadratic formula to find y in terms of x and got
    y = (-1(+/-)2^.5)x

    I don't know what to do from there. How do I find the critical points now that I have that equation for y?

    Thanks!
    Finding Critical Points:

    Find derivative of function, set the deriv equal to zero and solve for x.

    example:

    f(x) = x^3 - 3x^2 + 13
    f'(x) = 3x^2 - 6x

    3x^2 - 6x = 0
    3x(x-2) = 0
    x = 0, 2 <--These are your critical points.


    Plug the critical values 0 and 2 into the original equation.
    (0, 13) global max
    (2,9) global min
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  3. #3
    Junior Member
    Joined
    Nov 2011
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    Re: Finding global max and min of a function

    I know how to find the CP of a single-variable function... A multivariable function is different and more complex, and in the case I provided I am extremely confused as to what I am to do to find the actual coordinates of the critical points.
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