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Question

If x,y,z>0 and x+y+z=1, then the least value of 2x1x+2y1y+2z1z is

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Solution

We know that,
AMHM
Considering (x+y),(y+z) and (x+z) as three numbers, we get
(x+y)+(y+z)+(x+z)331x+y+1y+z+1x+z
2(x+y+z)(1x+y+1y+z+1x+z)9
1+zx+y+1+xy+z+1+yz+x92
x1x+y1y+z1z32
2x1x+2y1y+2z1z3

Hence, the least value of the given expression is 3.

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