Question

If $a,b$ and $c$ are positive real numbers, then $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ is greater than or equal to

• A. $3$
• B. $6$
• C. $27$
• D. None of these

Answer 1

The correct option is A $3$

Using A.M-G.M inequality,we have $A.M\ge G.M$

$\Rightarrow \frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{3}\ge {(\frac{a}{b}\times \frac{b}{c}\times \frac{c}{a})}^{\frac{1}{3}}$

$\Rightarrow \frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{3}\ge {\left(1\right)}^{\frac{1}{3}}$

$\Rightarrow \frac{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}}{3}\ge 1$

$\Rightarrow \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge 3$