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If a, b, c are positive real numbers th...

If a, b, c are positive real numbers then show that
(i) `(a + b + c) ((1)/(a) + (1)/(b) +(1)/(c)) ge 9`
(ii) `(b+c)/(a) +(c +a)/(b) + (a+b)/(c) ge 6`

Text Solution

Verified by Experts

(i) The inequality is straight forward if we use the fact that AM `ge` HM
` (a + b + c)/(3) ge(3)/((1)/(a) +(1)/(b) +(1)/(c))`
` rArr (a + b +c) ((1)/(a) +(1)/(b) +(1)/(c)) ge 9 `
` (a + b +c) ((1)/(a) +(1)/(b) +(1)/(c)) ge 9`
`(1 + (b+c)/(a)) + (1 +(c +a)/(b)) +(1+(a+b)/(c)) ge 9 `
Thus ` (b+c)/(a)+(c+a)/(b) +(a +b)/(c) ge 6`
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