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Relation between AM,GM,HM for 2 Numbers

The minimum v...

Question

The minimum value of $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c})$ for $a > 0, b > 0$ and $c > 0$ is

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Solution

Since, $A . M \geq H . M$
$∴ \frac{a + b + c}{3} \geq \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$
$\Rightarrow (a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$

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