Discussion
The functions :
\(f_{1}(x, y, z) = x + y + z\)
\(f_{2}(x, y, z) = xy + yz + zx\)
\(f_{3}(x, y, z) = x^{2} + y^{2} + z^{2}\)
Answer: D
The functions are functionally dependent on \(\mathbf{R}^{3}\).
They satisfy an identity everywhere:
\[ (x + y + z)^{2} = x^{2} + y^{2} + z^{2} + 2(xy + yz + zx) \]
that is, \(f_{1}^{2} - f_{3} - 2f_{2} = 0\). Equivalently the Jacobian \(\partial(f_{1}, f_{2}, f_{3})/\partial(x, y, z)\) vanishes identically: its rows are \((1, 1, 1)\), \((y + z, x + z, x + y)\) and \((2x, 2y, 2z)\), and the third row equals \(2f_{1}(1, 1, 1) - 2(y + z, x + z, x + y)\), so the rows are linearly dependent at every point.
Because the relation holds on all of \(\mathbf{R}^{3}\), the functions are not independent on any subset, so options A, B and C are wrong.
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