Answer
Maximum: $\dfrac{3}{2}$ and Minimum: $\dfrac{1}{2}$
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 as: $\nabla f(x,y,z)=\lambda_1 \nabla g(x,y,z)+\lambda_2 \nabla h(x,y,z)$
This yields $\nabla f=\lt a,z+x,y, \gt$ and $\lambda_1 \nabla g(x,y,z)+\lambda_2 \nabla h(x,y,z)= \lt ay,ax+2by,2bz \gt$
As per the given constraint condition we get, $x=\pm \sqrt 2$
Simplify. Thus, $xy=1$ and $y^2+z^2=1$ we get $z=\pm \dfrac{1}{\sqrt2}$
and $y=1-z^2=1-\pm \dfrac{1}{\sqrt2}=\pm \dfrac{1}{\sqrt2}$
The desired results are: Maximum: $\dfrac{3}{2}$ and Minimum: $\dfrac{1}{2}$
