Answer
$x=4,y=4,z=4$ and Minimum value: $x^2+y^2+z^2=48$
Work Step by Step
Lagrange Multipliers Method:
$F(x,y,z)=x^2+y^2+z^2, G(x,y,z)=x+y+z=12$
This gives $\lambda =2x$
$\lambda =2y$
$\lambda =2z$
Thus, $x=y=z=\dfrac{\lambda}{2}$
After solving for $x$, we get $x=4$
Now,
Hence, $x=4,y=4,z=4$ and Minimum value: $x^2+y^2+z^2=48$
