Hi,

Here is my question:

Find the point P on the parabola y=(x^2) closest to the point (3,0).

I solved it... but I am not sure what my domain restrictions should be...

Thanks,

D = sqrt( (x2 - x1)^2 + (y2 - y1)^2 )

Let (x1, y1) = (3, 0) and (x2, y2) = (x, y). Then

D = sqrt( (x - 3)^2 + (y - 0)^2 )

D = sqrt( (x - 3)^2 + y^2 )

*y = x^2

D = sqrt( (x - 3)^2 + (x^2)^2 )

D = sqrt( (x - 3)^2 + x^4 )

We want to minimize this by making dD/dx = 0.

We *could* differentiate this directly, but when we have to deal with derivatives with roots followed by the chain rule. I will instead square both sides, and then differentiate implicitly.

D^2 = (x - 3)^2 + x^4

Differentiate implicitly with respect to x,

2D (dD/dx) = 2(x - 3) + 4x^3

Make dD/dx = 0, to get

0 = 2(x - 3) + 4x^3

Solve for x.

0 = 2x - 6 + 4x^3

0 = 4x^3 + 2x - 6

0 = 2x^3 + x - 3

x=1... sub into original -> y=1. Therefore (1,1)