Given the surface xyz^2=5. Prove that on this surface there exists a point closest to the origin, and find that point.

Justify your solution.

Thanks alot!

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- May 26th 2008, 12:45 PMflgator11xyz^2=5 shortest distance
Given the surface xyz^2=5. Prove that on this surface there exists a point closest to the origin, and find that point.

Justify your solution.

Thanks alot! - May 26th 2008, 01:43 PMgalactus
You could try a LaGrange multiplier.

We want to find the extrema of subject to the constraint

Therefore, we have

Now, you can set [1] and [2] equal and solve for y. Set [1] and [3] equal and solve for z. Then sub them into the constraint to find your x value. Then you can sub that back into the other equations to find y and z.

Plug those into f(x,y,z) to find the point that is closest the origin.

There may be several to check. - May 27th 2008, 12:03 AMOpalg
In other words, we want to find the extrema of . Put the two partial derivatives equal to zero and you get the equations , . Thus . Discounting the solution x=0, you see that (with y=x and z=2x). I think that's easier than using Lagrange multipliers for this question.

- May 27th 2008, 04:31 AMgalactus
Yes, it is easier, but Lagrange is a good thing to learn. (Clapping)

- May 27th 2008, 11:34 AMflgator11
Well thanks to both of you for your help! It's much appreciated.