Question
If $$x, y, z$$ are three real numbers of the same sign then the value of $$\frac{x}{y} + \frac{y}{z} + \frac{z}{x}$$ lies in the interval
A.
$$\left[ {2, + \infty } \right)$$
B.
$$\left[ {3, + \infty } \right)$$
C.
$$\left( {3, + \infty } \right)$$
D.
$$\left( { - \infty ,3} \right)$$
Answer :
$$\left[ {3, + \infty } \right)$$
Solution :
$$\frac{x}{y},$$ e.t.c., are positive. $$A \geqslant G$$
$$ \Rightarrow \,\,\frac{{\frac{x}{y} + \frac{y}{z} + \frac{z}{x}}}{3} \geqslant \root 3 \of {\frac{x}{y} \cdot \frac{y}{z} \cdot \frac{z}{x}} = 1.$$