You have the 3 equations :

i 1 = 2ax + b

ii 0 = 4ay + b

iii 0 = 4az - b

adding ii and iii we get y = - z using this in the constraint x+y=z we

obtain:

x -z = z or x=2z

Using this in the other constraint x^2 +2y^2 +2z^2 = 8

We obtain:

4*z^2 + 2z^2 +2z^2 = 8

z=+1 , x=+2 , y = -1,1 (2,-1,1) and (-2,1,-1) are candidates for max and min

Max is 2 Min is -2 since f(x,y,z) = x