Minimum Value of the function of three variables using Lagrange Multipliers

Question

Find the minimum value of the function $f(x, y, z)=4x^2+2y^2+z^2$ with the constraint $g(x, y, z)=xy+yz+zx=16$.

I tried and get the following equations $8x=\lambda(y+z)$, $ 4y=\lambda (x+z) $, $2z=\lambda(x+y)$. From here how to solve the problem.

Answer 1

Now, solve the following system: $$\frac{8x}{y+z}=\frac{4y}{x+z}=\frac{2z}{x+y}$$ and $$xy+xz+yz=16.$$ For example, since $\frac{x}{y+z}=\frac{\lambda}{8},$ $\frac{y}{x+z}=\frac{\lambda}{4}$ and $\frac{z}{x+y}=\frac{\lambda}{2},$ we obtain: $$\frac{1}{1+\frac{\lambda}{8}}+\frac{1}{1+\frac{\lambda}{4}}+\frac{1}{1+\frac{\lambda}{2}}=2$$ or $$\lambda^3+7\lambda^2-32=0,$$ which gives a very ugly answer.