Question

find the points on the surface \[y^2 = 9 + xz\] that are closest to the origin

Answer 1

what class would this be for?
I have sort of an idea of what to do, but i just want to check to make sure im on the right track with this one... :P

Answer 2

It's calculus 3, from the chapter on maximum and minimum values

Answer 3

I'll try my best to help you identify what's going on, but I haven't actually done these myself. It does seem like I have the problem's idea right, tho.

Okay, so, do you have any ideas on how to start the problem? The key phrase I see in our question is "closest to the origin", which is suspicious in terms of minimum values.

Answer 4

I'm really lost when it comes to this one. The first thing that I did was graph it and there isn't a min or max anywhere near the origin

Answer 5

set up a distance formula between the surface and the origin and minimize it

Answer 6

this was on my multivar calc exam today

Answer 7

umm.. yeah, that's about what I was going for...
We're finding the closest point to the origin on our surface, so we'd be looking at the distance formula. lol

Answer 8

Okay, so something like this? \[\sqrt{(a-x)^2+(b-y)^2+(c-2)^2}\]

Answer 9

I'd write it like this since we're using the origin:and some point(x,y,z)
\[\begin{split} D &= \sqrt{(x - a)^2 + (y - b)^2 + (z - c)^2}~~~~~(a,b,c) = (0,0,0)\\
&=\sqrt{(x-0)^2 + (y-0)^2 + (z-0)^2}\\
&= \sqrt{x^2 + y^2 + z^2}\\ \end{split}\]

Answer 10

Going off of that, I think the next step would be to substitute \(y^2=9+xz\) into the distance formula.

Answer 11

okay, so \[\sqrt{x^2 + 9 +xz + z^2}\]

Answer 12

So that function definitely has a minimum over the origin

Answer 13

yeah, that's where I get to. I see examples online that prefer to square the distance and minimize the function squared since the square root isn't the most derivative-friendly function, but that seems a little beyond me. D:

Answer 14

Oh, that makes a lot of sense. If I square the function, it should still have a minimum at the same values of x and z

Answer 15

If you're interested in the imaginary roots things are slightly more complicated, but in this case it doesn't matter. The square of that function does still have a minimum at the same values.

Answer 16

Okay, I think that I got it. There is a local minimum of \[D = \sqrt{x^2+9+xz+z^2}\] at x = 0, z = 0

Answer 17

So I should be able to just plug those values into the original function which means that the points (0, 3, 0) and (0, -3, 0) are the closest?

Answer 18

yeah, that is what I am getting as well for the critical point, im just not sure how to check if its a minimum... :P

Answer 19

According to the almighty wolfram alpha, those points are correct.

Answer 20

Well cool, I think that I'm good then. They also check out visually with graphing software