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If a ,b ,c in R^+t h e n(a+b+c)(1/a+1/b...

If `a ,b ,c in R^+t h e n(a+b+c)(1/a+1/b+1/c)` is always `geq12` `geq9` `lt=12` none of these``

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Since, `a, b, c gt 0`
`rArr ((a +b+c))/(3) gt (abc)^(1//3)`...(i)
[using `AM ge GM`]
Also, `((1)/(a) + (1)/(b) + (1)/(c))/(3) ge ((1)/(a).(1)/(b).(1)/(c))^(1//3)`...(ii) [using `AM ge GM`]
On mulptiplying Eqs. (i) and (ii), we get
`((a +b +c)((1)/(a) + (1)/(b) + (1)/(c)))/(9) ge (abc)^(1//3) (1)/((abc)^(1//3))`
`:. (a + b + c) ((1)/(a) + (1)/(b) + (1)/(c)) ge 9`
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