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Question

If a,b,c are three distinct positive real numbers which are in H.P then 3a+2b2ab+3c+2b2cb is

A
Greater than or equal to 10
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B
Less than or equal to 10
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C
Only equal to 20
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D
None of these
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Solution

The correct option is A Greater than or equal to 10
a,b,c in H.P, b=2aca+c...(1)
3a+2b2ab+3c+2b2cb=6ac+4bc3ab2b2+6ac+4ab3bc2b24ac2ab2bc+b2
=12ac+b(a+c)4b24ac2b(a+c)+b2
Replace b(a+c)=2ac by relation (1)
=12ac+2ac4b24ac4ac+b2=14ac4b2b2
=14acb24
Replace 2acb=a+c,
7(a+c)b4
As a,b,c in H.P 2b=1a+1c
7(a+c)b4=7(a+c)12(1a+1c)4
=72(1+ac+ca+1)4
=72(2+ac+ca)4=3+72(ac+ca)
By Arithmetic mean relation on ac,ca
(ac+ca)2ac×ca
ac+ca2
3+72(ac+ca)3+72×2=10
Will be greater than or equal to 10.


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