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

Question:

If xyz are positive integers then value of the expression (x + y) (y + z) (z + x) is

(a) = 8xyz

(b) > 8xyz

(c) < 8xyz

(d) = 4xyz

Solution:

Let xyz be positive integers then (x + y) (y + z) (z + x) = ??

Since $\frac{x+y}{2}>\sqrt{x y}$ (Since arithmetic mean $\geq$ geometric mean)

i. e $x+y>2 \sqrt{x y}$

Similarly $y+z>2 \sqrt{y z}$ and $z+x>2 \sqrt{z x}$

$\therefore(x+y)(y+z)(z+x)>8 \sqrt{x y} \sqrt{y z} \sqrt{z x}$

$=8 \sqrt{x^{2} y^{2} z^{2}}$

$=8 x y z$

$\therefore(x+y)(y+z)(z+x)>8 x y z$

Hence, the correct answer is option B.

 

 

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