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Question

If the product of three positive real numbers is 1 and their sum is greater than sum of their reciprocals, then



A

exactly one of them exceeds 1

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B

each of them equals 1

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C

exactly one of them is less than 1 

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D

None of these

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Solution

The correct option is A

exactly one of them exceeds 1


Let the numbers be a, b, 1ab. We are given 
a+b+1ab>1a+1b+abNow,(a1)(b1)(1ab1)=1+(a+b+1ab)(ab+1a+1b)1>0
either all three of a-1, b-1, 1ab1 are positive or exactly one of them is positive.
But all three of a-1, b-1, 1ab1 cannot be positive.


Mathematics

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