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Find the minimum value of (a+b+c)(1/a+1/b+1/c)? This is a question from progressions ,Algebra.

Find the minimum value of (a+b+c)(1/a+1/b+1/c)?
This is a question from progressions ,Algebra.

Grade:11

1 Answers

Anoopam Mishra
126 Points
8 years ago
You must be knowing that AM \geqslant GM.
\frac{a+b+c}{3} \geqslant (abc)^{1/3} and \frac{1}{3} (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} )\geqslant (\frac{1}{a} \frac{1}{b} \frac{1}{c})^{1/3}
Multiplying both these inequalities, we get
=> \frac{1}{9}(a+b+c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} )\geqslant (abc)^{1/3} (\frac{1}{a} \frac{1}{b} \frac{1}{c})^{1/3}
=> (a+b+c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} )\geqslant 9
The minimum value of (a+b+c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c} ) is 9.

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