Find the greatest value of $x y z$ for positive value of $x, y, z$ subject to the condition $x y + y z + z x = 12.$

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Solution

Using the relation A.M. $\geq$ G.M.
$\frac{x y + y z + z x}{3} \geq {(x^{2} y^{2} z^{2})}^{1 / 3} \Rightarrow 4 \geq {(x y z)}^{2 / 3}$
$\Rightarrow$ $x y z \leq 8$

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