if x,y,z are distinct positive numbers ,prove that (x+y)(y+z)(z+x) 8xyz further if x+y+z=1 show that (1-x)(1-y)(1-z) 8xyz - Maths - Sequences and Series

Question

​if x,y,z are distinct positive numbers ,prove that
(x+y)(y+z)(z+x)>8xyz further if x+y+z=1 show that
(1-x)(1-y)(1-z)>8xyz

Rahul Raj · Accepted Answer

Dear student

For positive numbers, Arithmetic mean is always greater than or
equal to geometric mean

x+y2≥xy 1y+z2≥yz 2x+z2≥xz 3Multiplying the three
equations, we
getx+y2y+z2x+z2≥xyyzxz⇒x+yy+zz+x≥8xyzIf x+y+z=1,
then x+y=1-zz+y=1-xx+z=1-yNow, x+yy+zz+x≥8xyz⇒1-z1-x1-y
≥8xyz

Regards

​if x,y,z are distinct positive numbers ,prove that (x+y)(y+z)(z+x)>8xyz.further if x+y+z=1 show that (1-x)(1-y)(1-z)>8xyz

if x,y,z are distinct positive numbers ,prove that (x+y)(y+z)(z+x)>8xyz.further if x+y+z=1 show that (1-x)(1-y)(1-z)>8xyz