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Math Help - Finding point closest to origin

  1. #1
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    Finding point closest to origin

    I have finished this problem but I'm not quite sure if this has 2 answers or 1. Here is the question.

    Find the points on the parabola y= 9/2 - x^2 that are closest to the origin.

    Steps I did:

    Use the distance formula and got x^2 + (9/2 - x^2)^2

    S'= 4x(x^2 - 4)
    x = 0, x = +-2

    Points I got: (-2, 1/2), (2, 1/2)

    is there something I did wrong?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Finding point closest to origin

    For some arbitrary point on the parabola \left(x,\frac{9}{2}-x^2 \right), the square of the distance D from this point to the origin is:

    D^2=x^2+\left(\frac{9}{2}-x^2 \right)^2

    Implicitly differentiating. we find:

    D'(x)=\frac{2x(x+2)(x-2)}{\sqrt{x^2+\left(\frac{9}{2}-x^2 \right)^2}}

    Since the denominator has no real roots, our only critical values are:

    x=-2,\,0,\,2

    and the first derivative test shows that minima occur at x=\pm2. Since the distance function is even, we know the two points are the same distance from the origin, so we include them both.

    You did well!
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