Find the points on the level surface xy2z4=1 that are closest to the origin.
Lagrange's method for finding extrema
The Attempt at a Solution
If I have a level surface F(x,y,z)=c, it's points closest to the origin will be the ones in which the gradient vector points to the origin. A generic vector pointing to/from the origin is G=<x,y,z>, so F must be a scalar multiple of G.
I come up with a system of equations
I can first simplify a little bit.
I can set the 2nd and 3rd equations equal.
----> x= z2/2
I can plug that x into the first 2 equations.
----> y = +/- √(z4/2)
Plugging those into the constraint xy2z4=1
Am I right? What is the most straight-forward way of solving such a problem?