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Distance between point and curve

  1. Aug 13, 2010 #1
    I have tried both lagrange multiplier and basic derivative minimization for this but keep ending with an ugly polynomial. Any ideas would be appreciated:

    find the shortest distance between the curve <t, t^2> and (2,2)
  2. jcsd
  3. Aug 13, 2010 #2


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    Whether by Lagrange multipliers or direct substitution, I get [math]2x^3- 3x- 2= 0[/math]. Is that the "ugly" polynomial you mean? Yes, it has no rational roots. Probably the best you can do is use the cubic formula. Fortunately, it is alread in "reduced form"- there is no "x2" term. This is of the form x3+ mx= n with m= -3/2 and n= 1. A root is of the form a- b with
    [tex]a^3= \frac{n}{2}+ \sqrt{\left(\frac{n}{2}\right)^2+ \left(\frac{m}{3}\right)^2}[/tex]
    [tex]= \frac{1}{2}+ \sqrt{\frac{29}{8}}[/tex]
    [tex]b^3= -\frac{1}{2}+ \sqrt{\frac{29}{8}}[/tex]
  4. Aug 13, 2010 #3
    yeah this is what I got. Thanks for the reply, I just wanted to see if I was doing something wrong since I haven't typically had to use the cubic formula for textbook questions
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