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Math Help - Need helping starting - (sorry for bad thread name)

  1. #1
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    Need helping starting - (sorry for bad thread name)

    Hi,

    Sorry for the thread title i really didn't how to describe the question,

    I really need help starting this off - i've been heere for ages and i'm clearly missing a trick! Pulling hair almost!

    Question:

    A Point R has coordinates (x, y). Let D be the distance from (0, 0) to R so that

    D^2 = x^2 + y^2

    R Lies on the curve y = (1+ x^2)^-1

    Find the coordinates of R such that D^2 is a minimum.

    Thanks in advance if you can help!!
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  2. #2
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    Quote Originally Posted by aceband View Post
    Hi,

    Sorry for the thread title i really didn't how to describe the question,

    I really need help starting this off - i've been heere for ages and i'm clearly missing a trick! Pulling hair almost!

    Question:

    A Point R has coordinates (x, y). Let D be the distance from (0, 0) to R so that

    D^2 = x^2 + y^2

    R Lies on the curve y = (1+ x^2)^-1

    Find the coordinates of R such that D^2 is a minimum.

    Thanks in advance if you can help!!
    Plug in the term of y into the equation

    (D(x))^2 = x^2+\dfrac1{(1+x^2)^2}

    Differentiate (D(x)) and solve the equation for x:

    ((D(x)))' = 0

    I've got ((D(x))^2)' = 2x\cdot \left(1-\dfrac{2}{(x^2+1)^3}\right)

    This equation has 3 real solutions.

    You must check if the 3 corresponding points R actually produce a minimum at D.
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  3. #3
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    Thank you so much!

    Just to check, i'm getting these values for x which seem Very odd to me:

    <br />
x = 0 and x= \sqrt[2]{\sqrt[3]{2}-1}<br />

    :s
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  4. #4
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    Quote Originally Posted by aceband View Post
    Thank you so much!

    Just to check, i'm getting these values for x which seem Very odd to me:

    <br />
x = 0 ~\vee~ x= \sqrt[2]{\sqrt[3]{2}-1}<br />

    :s
    Your results are correct. I've got the points R as:

    R_1(0,1),\ R_2\left(\sqrt{\sqrt[3]{2} - 1},\ \frac32 \sqrt[3]{2} - 1\right),\ R_3\left(-\sqrt{\sqrt[3]{2} - 1},\ \frac32 \sqrt[3]{2} - 1\right)


    Btw:
    1. There are 3 x-values.
    2. The solutions are connected by an or-sign.
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