# Finding point closest to origin

• Nov 7th 2012, 09:03 PM
illicitkush
Finding point closest to origin
I have finished this problem but I'm not quite sure if this has 2 answers or 1. Here is the question.

Find the points on the parabola y= 9/2 - x^2 that are closest to the origin.

Steps I did:

Use the distance formula and got x^2 + (9/2 - x^2)^2

S'= 4x(x^2 - 4)
x = 0, x = +-2

Points I got: (-2, 1/2), (2, 1/2)

is there something I did wrong?(Thinking)
• Nov 7th 2012, 09:24 PM
MarkFL
Re: Finding point closest to origin
For some arbitrary point on the parabola $\displaystyle \left(x,\frac{9}{2}-x^2 \right)$, the square of the distance $\displaystyle D$ from this point to the origin is:

$\displaystyle D^2=x^2+\left(\frac{9}{2}-x^2 \right)^2$

Implicitly differentiating. we find:

$\displaystyle D'(x)=\frac{2x(x+2)(x-2)}{\sqrt{x^2+\left(\frac{9}{2}-x^2 \right)^2}}$

Since the denominator has no real roots, our only critical values are:

$\displaystyle x=-2,\,0,\,2$

and the first derivative test shows that minima occur at $\displaystyle x=\pm2$. Since the distance function is even, we know the two points are the same distance from the origin, so we include them both.

You did well! :)