Find the Lagrange multipliers $f(x,y,z)=x+xyz$ subject to constraint $g(x,y,z)=x/3+y/2+z/4 - c= 0$
Criteria: $x,y,z,c$ are positive integers, c is constant, and the constraints are $x≥0$, $y≥5$, $z≥50$, $c≥0$.
My attempt: $$\langle f_x,f_y,f_z\rangle = λ\langle g_x,g_y,g_z\rangle$$ $$\langle 1+yz,xz,xy\rangle = λ\langle 1/3,1/2,1/4\rangle$$ $$3+3yz = 2xz = 4xy$$ $$2xz = 4xy \implies x = 0 \text{ or } z=2y$$
For $z=2y$, $$3+3yz = 2xz$$ $$3+3y(2y) = 2x(2y)$$ $$6y^2 -4xy +3 = 0$$
What's next?