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(C) We use the Arithmetic Mean-Harmonic Mean inequality $(AM \geq HM)$.For three positive numbers $a, b, c$,the Arithmetic Mean is $\frac{a+b+c}{3}$ a

Vedclass · Accepted Answer

$9$

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(C) We use the Arithmetic Mean-Harmonic Mean inequality $(AM \geq HM)$.For three positive numbers $a, b, c$,the Arithmetic Mean is $\frac{a+b+c}{3}$ and the Harmonic Mean is $\frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$.According to the inequality,$\frac{a+b+c}{3} \geq \frac{3}{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}$.Multiplying both sides by $3 \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} ight)$,we get $(a+b+c) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} ight) \geq 3 	imes 3 = 9$.Thus,the minimum value is $9$.

If $a, b, c$ are any three positive numbers,then what is the minimum value of $(a + b + c) \left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \right)$?

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