If x , y , z are real and distinct, then x 2 + 4 y 2 + 9 z 2 - 6 y z - 3 z x - 2 x y is always

Question

If $x, y, z$ are real and distinct, then $x^{2} + 4 y^{2} + 9 z^{2} - 6 y z - 3 z x - 2 x y$ is always

Smart Achievers · Accepted Answer

(1)

$f (x, y, z) = x^{2} + 4 y^{2} + 9 z^{2} - 6 y z - 3 x z - 2 x y$

$= \frac{1}{2} [2 x^{2} + 8 y^{2} + 18 z^{2} - 12 y z - 6 x z - 4 x y]$

$= \frac{1}{2} [{(x - 2 y)}^{2} + {(2 y - 3 z)}^{2} + {(3 z - x)}^{2}] \geq 0$

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