If a b c are positive and a+b+c=1 then the least value of 1/a+1/b+1/c is
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Assuming that a, b, c are positive.
It is a well know fact that A.M≥H.M.A.M≥H.M.
Applying this to a,b,ca,b,c
a+b+c3≥3(1a+1b+1c)a+b+c3≥3(1a+1b+1c)
Cross multiplying gives that
(a+b+c)(1a+1b+1c)≥9(a+b+c)(1a+1b+1c)≥9
Thus, the minimum value is 99, when a=b=c=1
It is a well know fact that A.M≥H.M.A.M≥H.M.
Applying this to a,b,ca,b,c
a+b+c3≥3(1a+1b+1c)a+b+c3≥3(1a+1b+1c)
Cross multiplying gives that
(a+b+c)(1a+1b+1c)≥9(a+b+c)(1a+1b+1c)≥9
Thus, the minimum value is 99, when a=b=c=1
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