Show that $f (x, y, z) = a x^{2} + a y^{2} + a z^{2} + 2 b x y z$ is Symmetric.

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Solution

Consider $f (x, y, z) = a x^{2} + a y^{2} + a z^{2} + 2 b x y z$ Observe that $f (x, y, z) = f (y, x, z)$
(x and y are interchanged)
$f (x, y, z) = f (x, z, y)$
(y and z are interchanged) $f (x, y, z) = f (z, y, x)$
(x and z are interchanged)
Thus $f (x, y, z)$ is symmetric with respect to any two variables of x, y and z.

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