Answer
Minimum:$f(4,4,4)=48$. Maximum value does not exist.
Work Step by Step
Use Lagrange Multipliers Method:
$\nabla f(x,y,z)=\lambda \nabla g(x,y,z)$
This yields $\nabla f=\lt 2x,2y,2z \gt$ and $\lambda \nabla g=\lambda \lt 1,1,1\gt$
Using the constraint condition we get, $2x=\lambda, 2y=\lambda,2z=\lambda $
After solving, we get $x=y=z$
Since, $g(x,y)=x+y+z=12$ gives $x=\lambda/12$
Thus, $\lambda=8$, so $x=y=z=4$
Hence,
Minimum:$f(4,4,4)=48$. Maximum value does not exist.
