If a- b- c - 0 prove a-b-c1a-1b-1c - 9-

since AM≥ GM≥ HM, AM≥ HM a+b+c / 3 ≥ 3 ( 1 a + 1 b + 1 c ) (a+b+c)( 1 a + 1 b + 1 c )≥ 9 ...

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Quadratic Equations

If a, b, c ...

Question

If a, b, c $> 0$ prove $(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c}) \geq 9$ .

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a (\frac{1}{b} + \frac{1}{c}), b (\frac{1}{c} + \frac{1}{a}), c (\frac{1}{a} + \frac{1}{b})

a, b, c

{(\frac{1}{x^{a - b}})}^{\frac{1}{a - c}} {(x^{\frac{1}{b - c}})}^{\frac{1}{b - a}} {(x^{\frac{1}{c - a}})}^{\frac{1}{c - b}} = 1

(a + b + c) (\frac{1}{a} + \frac{1}{b} + \frac{1}{c})

a > 0, b > 0

c > 0

> 0

(a b c) {(\frac{1}{a} + \frac{1}{b} + \frac{1}{c})}^{3} \geq 27

A (\frac{2 c}{a}, \frac{c}{b}), B (\frac{c}{a}, 0)

C (\frac{1 + c}{a}, \frac{1}{b})

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