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Math Help - Find a tangent plane horizontal to the xy plane?

  1. #1
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    Find a tangent plane horizontal to the xy plane?

    find the points on the surface f(x,y,z)=1 that have a horizontal plane (ie a tangent plane horizontal to the xy plane)
    where
    f(x,y,z)= x^3 +y^34 -z^3.

    I'm getting stuck as I can only find one point.

    I got (0,0,-1) but this is by using a random vector I set as (0,0,-3z^2).
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  2. #2
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    Quote Originally Posted by Roland25 View Post
    find the points on the surface f(x,y,z)=1 that have a horizontal plane (ie a tangent plane horizontal to the xy plane)
    where
    f(x,y,z)= x^3 +y^34 -z^3.

    I'm getting stuck as I can only find one point.

    I got (0,0,-1) but this is by using a random vector I set as (0,0,-3z^2).
    Surely you don't mean "horizontal to the xy plane". Either just "horizontal" or "parallel to the xy plane". Any such plane is of the form z= constant and a unit normal vector is <0, 0, 1>. A vector normal to the surface given by f(x,y,z)= x^2+ y^3- z^3 is \nabla f= <2x, 3y^2, -3z^2>. That will be parallel to <0, 0, 1> when 2x= 0, 3y^2= 0.

    (Surely you didn't mean " y^{34}". I have assumed it was " y^3 and you accidently hit two keys.)
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  3. #3
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    Sorry, you were right, it wasm eant to be y^4.

    The question states find the points on the surface that have a horizontal tangent plane
    which I'm guessing means a tangent plane parallel to the xyplane right?
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