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Tuesday, September 16, 2008

Q) Prove that if abc=1 and a,b,c are positive real numbers.

A) Rewrite the expression as
(a-1 + 1/b + b - b) (b - 1 + 1/c + c - c) (c - 1 + 1/a + a - a)
Max value of the terms will be obtained when 1/b + b, 1/c+ c and 1/a +a are max, since a, b , c are all positive.
And we know that 1/n + n >= 2
(and for max value, n = 1)
So a=b=c=1 for the max value :

(a -b +1)(b- c+1)(c - a+1)
= 1*1*1
= max value --> 1
so the expression is less than or equal to 1.


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