Question: Use the Lagrange multiplier method to find the extrema of \(f\) subject to the given constraints:

1. \(f(x, y, z)=x-y+z, \quad x^{2}+y^{2}+z^{2}=2 ;\)
2. \(f(x, y)=x-y, \quad x^{2}-y^{2}=2 ;\)
3. \(f(x, y, z)=x+y+z, \quad x^{2}-y^{2}=1,2 x+z=1\).

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