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    Hey, I've been asked to find the stationary points of the function

    f(x,y) = (1/3)x^3 + (1/3)y^3 - 4xy^2 + 15y = 0

    I have found the partial derivatives.

    df/dx = x^2 - 4y^2

    df/dy = y^2 - 8xy + 15

    I know that the stationary points occure when df/dx = df/dy = 0

    how do I solve these derivates? I tried rearranging it but that didn't work.
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    (Original post by sarah_vickers)
    Hey, I've been asked to find the stationary points of the function

    f(x,y) = (1/3)x^3 + (1/3)y^3 - 4xy^2 + 15y = 0

    I have found the partial derivatives.

    df/dx = x^2 - 4y^2

    df/dy = y^2 - 8xy + 15

    I know that the stationary points occure when df/dx = df/dy = 0

    how do I solve these derivates? I tried rearranging it but that didn't work.
    From the first equation x=\pm 2y
    Substitute it into the second
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    (Original post by ztibor)
    From the first equation x=\pm 2y
    Substitute it into the second
    when substituting x = 2y into df/dy I get y = +/- 1

    when substituting x = -2y into df/dy I get y = sqrt(-15/17) which cannot be defined.

    so my stationary points are just

    (2y,1), (2y,-1), (-2y,1) and (-2y,-1) which will give me (2,1), (-2,-1), (-2,1) and (2,-1)

    is that correct?
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    (Original post by sarah_vickers)
    when substituting x = 2y into df/dy I get y = +/- 1

    when substituting x = -2y into df/dy I get y = sqrt(-15/17) which cannot be defined.

    so my stationary points are just

    (2y,1), (2y,-1), (-2y,1) and (-2y,-1) which will give me (2,1), (-2,-1), (-2,1) and (2,-1)

    is that correct?
    No. AS you wrote the last two are not solutions
    as for both points x=-2y
    and if x=-2y then y will complex value and will not 1 or -1.
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    (Original post by ztibor)
    No. AS you wrote the last two are not solutions
    as for both points x=-2y
    and if x=-2y then y will complex value and will not 1 or -1.
    so my stationary points are just (2,1) and (-2,-1)?
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    how do you find the nature of the turning points? max or min or saddle?
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    my teacher has clarified that (2,1) and (-2,-1) are the stationary points.

    I have found the nature of the stationary points to both be saddlepoints. Is the nature correct? My teacher has not talked through this yet.
 
 
 
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