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Math Help - Bonus question involving derivatives?

  1. #1
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    Bonus question involving derivatives?

    Show that the point on the graph of y=lnx which is closest to the origin lies on the parabola y= -x^2.Draw a sketch. Also, find an approximate value of the x-coordinate of that point (with 4 decimals).

    I have no clue how to approach this problem, can anyone guide me through this question?

    thx in advance!
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  2. #2
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    Quote Originally Posted by Solid8Snake View Post
    Show that the point on the graph of y=lnx which is closest to the origin lies on the parabola y= -x^2.Draw a sketch. Also, find an approximate value of the x-coordinate of that point (with 4 decimals).

    I have no clue how to approach this problem, can anyone guide me through this question?

    thx in advance!
    You want to know the distance between any point on the graph of y = \ln{x} and the origin. Call any point on the graph (x, y) = (x, \ln{x}).

    Then the distance is

    D = \sqrt{(x - 0)^2 + (\ln{x} - 0)^2}

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


    You want to minimise this distance, so differentiate, set it equal to 0, solve for x and check using the second derivative test that it is in fact a minimum.
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