Finding point closest to origin

For some arbitrary point on the parabola $\left(x,\frac{9}{2}-x^2 \right)$ , the square of the distance $D$ from this point to the origin is:

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

Implicitly differentiating. we find:

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

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

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

and the first derivative test shows that minima occur at $x=\pm2$ . 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! :)