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Math Help - Point on curve thats closest to the origin

  1. #1
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    Point on curve thats closest to the origin

    Find the point on the curve y=sqrt(1/2ln(1/x)) that is closest to the origin

    I have attempted to do this question a couple of times, but I can't get the right answer. Please help.
    Thanks in advance.
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  2. #2
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    Re: Point on curve thats closest to the origin

    D^2 = (x - 0)^2 + (y - 0)^2

    let D^2 = u ... minimizing u will minimize D^2

    u = x^2 + \frac{1}{2} \ln\left(\frac{1}{x}\right)

    u = x^2 - \frac{1}{2} \ln{x}

    \frac{du}{dx} = 2x - \frac{1}{2x} = 0

    4x^2 - 1 = 0

    x = \frac{1}{2}

    \frac{d^2u}{dx^2} = 2 + \frac{1}{2x^2} > 0 \implies critical value is a minimum

    I'll leave you to find the y-coordinate of the point.
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