If `x+y+z=1a n dx ,y ,z` are positive, then show that `(x+1/x)^2+(y+1/y)^2+(z+1/z)^2>(100)/3`

Explanation

A.M of 2nd power `gt` 2nd power of A.M
`implies ((x + (1)/(2))^(2) + (y + (1)/(y))^(2) + (z + (1)/(z))^(2))/(3)`
`gt [((x + (1)/(2)) + (y + (1)/(y)) + (z + (1)/(z)))/(3)]^(2)`
or `((x + (1)/(2))^(2) + (y + (1)/(y))^(2) + (z + (1)/(z))^(2))/(3)`
`gt (1)/(9) (x + y + z + (1)/(x) + (1)/(y) + (1)/(z))^(2)`
or `((x + (1)/(2))^(2) + (y + (1)/(y))^(2) + (z + (1)/(z))^(2))/(3) gt (1)/(9) (1 + (1)/(x) + (1)/(y) + (1)/(z))^(2)`
Again `(x^(-1) + y^(-1) + z^(-1))/(3) gt ((x + + z)/(3))^(-1)`