Question

If $x, y, z$ are positive integers, then value of expression $(x + y)(y + z)(z + x)$ is

• A. $\ne 8xyz$
• B. $\ge 8xyz$
• C. $\le 8 xyz$
• D. $4xyz$

Answer 1

Answer: (B)

We know that, $A.M. \ge G.M$.
$\Rightarrow \frac{x+y}{2} \ge \sqrt{xy}, \frac{y+z}{2} \ge \sqrt{yz}$ and
$\frac{z+x}{2} \ge\sqrt{zx}$
Multiplying the three inequalities, we get
$ \frac{x+y}{2}\cdot \frac{y+z}{2}\cdot\frac{ x+z}{2} \ge\sqrt{ \left(xy\right)\left(yz\right)\left(zx\right)}$
or, $\left(x+y\right)\left(y+z\right)\left(z+x\right)\ge 8xyz$