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!
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.