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.