The Method of Langrange multipliers

The problem gives you the objection function f(x,y,z)=1+xy+yz subject
to the constraint x^2+y^2+z^2 =<1 (a unit sphere) and asks to find the
maximum.I fallow the procedure and obtain x=z=±1/2 and y=±sqrt(2)/2.
Since we are looking for max I took the positive values only. However
I do get a different answer from what I supposed to get. What do I do
wrong?
.

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