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Math Help - Help with Critical Points

  1. #1
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    Help with Critical Points

    Hey could someone take me through the steps of finding the critical points of the following function:

    f(x,y) = (x^2y^4)/4 + 4x^2 - xy^3 + y^2

    Thankyou!
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  2. #2
    MHF Contributor

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    The critical points of a function are the points at which the partial derivatives are 0 or the derivatives do not exist.

    This function, f(x,y) = (x^2y^4)/4 + 4x^2 - xy^3 + y^2 , is a polynomial in x and y so it can be done using the fact that the derivative of x^n, with respect to x, is nx^{n-1} and the derivative of y^n, with respect to y, is ny^{n-1}. Where are you having difficulty? Have you found the partial derivative functions?
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  3. #3
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    Yes so i have managed to do this

    fx = (xy^4)/2 + 8x - y^2

    0 = x(1/2y^4 + 8 - y^3) .... 1.

    fy = x^2 - 3xy^2 +2y

    0 = y(x^2y^2 - 3xy +2) ...... 2.

    Eqn 1. x = 0 or 1/2y^4 + 8 - y^3 = 0

    with x = 0 Eqn 2. becomes 2y = 0 so y = 0
    therefore first critical point = (0,0)

    with 1/2y^4 + 8 - y^3 = 0

    And now i'm stuck here. Have I done the right thing so far? I am really confused because i'm trying to following an example in my text book but its a much simpler function than my own.
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