Answer
Maximum: $f(2,1,0)=5$ and Minimum: $f(0,-1,0)=1$
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:$\nabla f(x,y)=\lambda \nabla g(x,y)$
Re-write in 3D as: $\nabla f(x,y,z)=\lambda_1 \nabla g(x,y,z)+\lambda_2 \nabla h(x,y,z)$
This yields $\nabla f=\lt 2x,2y,2z \gt$ and $\lambda_1 \nabla g(x,y,z)+\lambda_2 \nabla h(x,y,z)= \lt \lambda_1,-\lambda_1+2\lambda_2y,-2\lambda_2 z \gt$
As per the given constraint condition $x-y=1$we get, $x=2/3 ,y=-1/3$
We can see that when we use the values $x=2/3 ,y=-1/3$ of $x,y$ in another constraint $y^2-z^2=1$, the invalid value will be achieved.
In order to get the valid or value need to consider $z=0$; after simplifications, we have $x=2,0$ and $y=1,-1$
Thus, Maximum value is $f(2,1,0)=5$ and Minimum value is $f(0,-1,0)=1$
