If `x ,y ,z`are distinct positive numbers, then prove that `(x+y)(y+z) – InterviewSolution

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1.	If `x ,y ,z`are distinct positive numbers, then prove that `(x+y)(y+z)(z+x)>8x y zdot`
Answer» Using `AM gt GM`, we have `(x+y)/2 gt sqrt(xy), (y+z)/2 gt sqrt(yz)` and `(z+x)/2 gt sqrt(zx)` `rArr x+y gt 2sqrt(xy), y+z gt 2sqrt(yz) and z+x gt 2sqrt(zx)` `rArr (x+y) (y+z) (z+x) gt 2 sqrt(xy)xx2 sqrt(yz)xx 2sqrt(zx)` `rArr (x+y)(y+z)(z+x) gt 8xyz`. Hence, `(x+y)(y+z)(z+x) gt 8xyz`.

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