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Math Help - Horizontal Tangent Plane

  1. #1
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    Horizontal Tangent Plane

    At which points of the surface x^3y-xyz+z^2=16 is the tangent plane horizontal?

    I made a function F(x,y,z)=x^3y-xyz+z^2-16 and set the partials to x and y as zero. I found three points this way: (0,0,4), (0,0,-4). and (2,0,4). However, the answer key states that (-2,0,4) is also a point. For me, this point didn't work out due to getting an imaginary root. I would much appreciate if someone would show me how to get this last point.
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  2. #2
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    Quote Originally Posted by MathTooHard View Post
    At which points of the surface x^3y-xyz+z^2=16 is the tangent plane horizontal?

    I made a function F(x,y,z)=x^3y-xyz+z^2-16 and set the partials to x and y as zero. I found three points this way: (0,0,4), (0,0,-4). and (2,0,4). However, the answer key states that (-2,0,4) is also a point. For me, this point didn't work out due to getting an imaginary root. I would much appreciate if someone would show me how to get this last point.
    You require the simultaneous solution to:

    3x^2y - yz = 0 .... (1)

    x^3 - xz = 0 .... (2)

    (-2, 0, 4) is clearly a solution.


    Note: x^4 = 16 \Rightarrow x = \pm 2 ....
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