Question

If x, y, z are positive integers then value of the expression (x + y) (y + z) (z + x) is
(a) = 8xyz
(b) > 8xyz
(c) < 8xyz
(d) = 4xyz

Answer 1

Let x, y, z be positive integers then (x + y) (y + z) (z + x) = ??
Since $\frac{x+y}{2}>\sqrt{xy}$ (Since arithmetic mean ≥ geometric mean)
$$\begin{gathered} \mathrm{i}.\mathrm{e}x+y>2\sqrt{xy} \\ \mathrm{Similarly}y+z>2\sqrt{yz}\mathrm{and}z+x>2\sqrt{zx} \end{gathered}$$
$\begin{array}{rcl}\therefore \left(x+y\right)\left(y+z\right)\left(z+x\right)& >& 8\sqrt{xy}\sqrt{yz}\sqrt{zx}\\ & =& 8\sqrt{x^2{y}^2{z}^2}\\ & =& 8xyz\\ \therefore \left(x+y\right)\left(y+z\right)\left(z+x\right)& >& 8xyz\\ & & \end{array}$

Hence, the correct answer is option B.