Given a,b and c are real numbers and a<b<c. If (1/a) + (1/b) + (1/c) = (1/(18)) , find minimum value of a.


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Question Number 134069 by john_santu last updated on 27/Feb/21

$G i v e n a, b a n d c a r e r e a l n u m b e r s a n d a < b < c .$ $I f \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{18}, f i n d m i n i m u m v a l u e o f a .$
Commented by mr W last updated on 27/Feb/21

$n o m i n i m u m f o r a e x i s t s .$
Commented by mr W last updated on 27/Feb/21

$a = \frac{1}{\frac{1}{18} - (\frac{1}{b} + \frac{1}{c})} \to 18 i f b, c \to + \infty$ $a = \frac{1}{\frac{1}{18} - (\frac{1}{b} + \frac{1}{c})} \to - \infty i f b, c \to 0^{+}$
	Answered by bramlexs22 last updated on 27/Feb/21

	$AM - GM$ $\frac{\frac{1}{a} + \frac{1}{b} + \frac{1}{c}}{3} ⩾ \sqrt[3]{\frac{1}{abc}}$ $\frac{[\frac{1}{18}]}{3} ⩾ \sqrt[3]{\frac{1}{abc}} \Rightarrow \frac{1}{54} ⩾ \sqrt[3]{\frac{1}{abc}}$ $\frac{1}{54^{3}} ⩾ \frac{1}{abc}, it hold for a = b = c$ $we get a = 54$
	Commented by mr W last updated on 27/Feb/21

	$w h a t d o y o u g e t i f b = c = 1000000 ?$ $w h a t d o y o u g e t i f b = c = 1 / 1000000 ?$
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