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Math Help - z=xy+1 shortest distance

  1. #1
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    z=xy+1 shortest distance

    Given the surface z=xy+1. Prove that on this surface there exists a point closest to the origin, and find that point.

    Justify your solution.

    Thank you very much!
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  2. #2
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    Quote Originally Posted by littlemu View Post
    Given the surface z=xy+1. Prove that on this surface there exists a point closest to the origin, and find that point.

    Justify your solution.

    Thank you very much!
    Since z=xy+1, thus any point on surface is (x,y,xy+1)

    The distance of this point from the origin is,
    sqrt(x^2+y^2+(xy+1)^2) = sqrt(x^2+y^2+x^2y^2+2xy+1)
    We need to minimize this.
    Instead let us minize its square, that is to remove the square root.

    We need to minimize,
    g(x,y)=x^2+y^2+x^2y^2+2xy+1

    Find its partial derivatives and make them zero.
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