If x1,x2,x3,x4 are four positive real numbers such that x1+1x2=4,x2+1x3=1,x3+1x4=4 and x4+1x1=1, then
A
x1=x3
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B
x2=x4
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C
x1x2=1
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D
x3x4=1
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Solution
The correct option is Dx3x4=1 Usingx+1y≥2√xy,weget4=x1+1x2≥2√x1x21=x2+1x3≥2√x2x34=x3+1x4≥2√x3x41=x4+1x1≥2√x4x1⇒16=(x1+1x2)(x2+1x3)(x3+1x4)(x4+1x1)≥24
Hence, each of the above inequality must be an equality.
Therefore, x1=1x2,x2=1x3,x3=1x4andx4=1x1.∴x1=2,x2=12,x3=2,x4=12