"Find the point on the parabola 
that is closest to the point )
"
:S what the fuck do you do with this..? tangents?
Let
)
be the points that are closest to
)
.
As Glockmeister said we need to use the distance formula. The distance between those two points are:
^2+(y-0)^2})
Since we know
we can sub this in:
^2+(x^2)^2})
^2+x^4})
We need to find when the distance is minimal, i.e. when
. We can also just ignore the square root as we are going to make
(since if we derive
then
^2+x^4\right)^{-\frac{1}{2}}(4x^3+2x-6))
)
We need to factorise this to find solutions. Let
=4x^3+2x-6)
, then:
=4(1)^3+2(1)-6=0)
Hence
.
We also need to ensure that this stationary point is a minimum. Find a point before and after
and substitute into
to verify the nature.
=-6)
=30)
Drawing a gradient table: \ _ / hence minimum. (Also by
^2+2=14>0)
therefore minimum).
Substitute
into
:
Therefore the point closest to
on the parabola
is
)
.
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