Answer
Maximum: $\dfrac{3}{2}$ and Minimum: $\dfrac{1}{2}$
Work Step by Step
Use Lagrange Multipliers Method:
$\nabla f(x,y,z)=\lambda_1 \nabla g(x,y,z)+\lambda_2 \nabla h(x,y,z)$
This yields $\nabla f=\lt a,z+x,y, \gt$ and $\lambda_1 \nabla g(x,y,z)+\lambda_2 \nabla h(x,y,z)= \lt ay,ax+2by,2bz \gt$
Using the constraint condition we get, $x=\pm \sqrt 2$
After solving, $xy=1$ and $y^2+z^2=1$ we get $z=\pm \dfrac{1}{\sqrt2}$
and $y=1-z^2=1-\pm \dfrac{1}{\sqrt2}=\pm \dfrac{1}{\sqrt2}$
Hence, Maximum: $\dfrac{3}{2}$ and Minimum: $\dfrac{1}{2}$
