Given-a-b-and-c-are-real-numbers-and-a-lt-b-lt-c-If-1-a-1-b-1-c-1-18-find-minimum-value-of-a-

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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