Answer
Maximum:$f(2,2,1)=9$, Minimum: $f(-2-2,-1)=-9$
Work Step by Step
Our aim is to calculate the extreme values with the help of Lagrange Multipliers Method subject to the given constraints. For this, we have:$\nabla f(x,y)=\lambda \nabla g(x,y)$
This yields $\nabla f=\lt 2,2,1 \gt$ and $\lambda \nabla g=\lambda \lt 2x,2y,2z\gt$
From the given constraint condition $x^2+y^2+z^2=9$ we get, $2=\lambda 2x, 2=\lambda 2y,1=\lambda 2z$
Simplify. Thus, $x=y=\pm 2$
Since, $g(x,y)=x^2+y^2+z^2=9$ $\implies$ $z=\pm 1$
Hence, Maximum value is $f(2,2,1)=9$, Minimum value is $f(-2-2,-1)=-9$
