TITLE: Minimizing a quadratic function subject to quadratic constraints
QUESTION [11 upvotes]: Okay, so I am attempting to minimize the function 
$$f(x,y, z) = x^2 + y^2 + z^2$$
subject to the constraint of
$$4x^2 + 2y^2 +z^2 = 4$$
I attempted to solve using Lagrange multiplier method, but was unable to find a $\lambda$ that made the system consistent.
$$2x = \lambda8x$$
$$2y = \lambda4y$$
$$2z = \lambda2z$$
Wouldn't it be the case that the first equation suggests $\lambda = 1/4$ but the second equation suggests $\lambda = 1/2$? I am unsure of where to go from here although I have spent time trying to figure out. 
Intuitively, I know that it must be the case that the minimum occurs at $(1,0,0)$ and is equal to $1$, but I can not show this using mathematical reasoning.

REPLY [3 votes]: To complement the other answers, we are looking for the points on a given ellipsoid that are closest to the origin. Here is the plot a sphere of radius $1$ touching the given ellipsoid at $(\pm 1, 0, 0)$.