a) write down the equations to find the point on the surface (x+y)z^2=1 in the first octant, closest to the origin, using the method of Lagrange multipliers.
b) solve these equations to find the closest point
the following should provide a good starting point.
math.stackexchange.com/questions/2179646/using-lagrange-multipliers-to-determine-the-point-on-a-surface-nearest-to-p