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Question

If x,y,zR+ then xyx+y+yzy+z+xzx+z is always


A

2(x+y+z)

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B

12(x+y+z)

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C

4(x+y+z)

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D

None of the above

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Solution

The correct option is B

12(x+y+z)


From A.M. G.M. H.M.

We know that A.M. H.M. and G.M. H.M.

x+y22xyx+y
xyx+y14(x+y)(1)
y+z22yzy+z
yzy+z14(y+z)(2)

Similarly, xzx+z14(x+z)(3)

Adding (1), (2) and (3),

xyx+y+xyx+y+xzx+z14(2x+2y+2z)

xyx+y+xyx+y+xzx+z12(x+y+z)

Hence, Option b. is true


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