Question

Let x,y,z be positive reals , which of the which following imples `x=y=z?`
A. `x^(3) +y^(3) +z^(3) =3xyz`
B. `x^(3) +y^(3)z +yz^(2) =3xyz`
C. `x^(3) +y^(3)z +z^(2)x =3xyz`
D. `(x+y+z)^(3) =27xyz`

Answer 1

Correct Answer - B
`(i) (x^(3) +y^(3) +z^(3))/(3) =(x^(3) y^(3) z^(3) )^(1//3)`
hence x=y=z { AM=GM}
(ii) `(x^(3) +y^(2) z+yz^(2) )/(3) le (x^(3) y^(3) z^(3) ) ^(1//3)(AM=GM)`
`(iii) x^(3) +y^(2) z+x=3xuyz`
`(x^(3) +(y^(2) z)/(2) +(y^(2)z)/(2) +z^(2) x)/(4) ge((x^(4) y^(4) z^(4) )/(4))^(1//4)`
`implies (3xyz)/(4) ge((xyz))/(sqrt(2))`Not possible
`(iv) (x+y+z)/(3) ge (xyz)^(1//3) implies (x+y+z) ^(1//3) implies (x+y+z)^(3)=27xyz.`