Distance between point and curve

[ocohen]

hi,
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)

 

[HallsofIvy]

Whether by Lagrange multipliers or direct substitution, I get \(\displaystyle 2x^3- 3x- 2= 0\). 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]
and
[tex]b^3= -\frac{1}{2}+ \sqrt{\frac{29}{8}}[/tex]

 

[ocohen]

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