Answer
$(4,4,4)$ and $48$
Work Step by Step
From the given statement let us consider
$F(x,y,z)=x^2+y^2+z^2, G(x,y,z)=x+y+z=12$
Need to apply Lagrange Multipliers Method to determine the three positive numbers.
we have $\nabla f=\lambda \nabla g$
$F(x,y,z)=x^2+y^2+z^2, G(x,y,z)=x+y+z=12$
or, $\lambda =2x$
and $\lambda =2y$
and $\lambda =2z$
or, $x=y=z=\dfrac{\lambda}{2}$
Simplify to get value for $x$.
we get $x=4$
Therefore, $x=4,y=4,z=4$;
Minimum value: $x^2+y^2+z^2=(4)^2+(4)^2+(4)^2=48$
