Thanks for the query.
As per C.S. Inequality -
(x.1 + y.1 + z.1)^2 < = (x^2 + y^2 + z^2)(1^2 + 1^2 + 1^2) = 3(x^2 + y^2 + z^2) ................... (1)
Again, applying C.S. Inequality on x^1/2, y^1/2 and z^1/2 we get,
(x^2 + y^2 + z^2)^2 < = (x^3 + y^3 + z^3)(x + y + z) = 81(x + y + z) .........................................(2)
Combining (1) and (2) we get,
(x + y + z)^4 < = 9(x^2 + y^2 + z^2)^2 which is again < = 9*81(x + y + z)
Therefore (x + y + z)^3 < = 9*81
or, x + y + z is less than equal to 9
Q.E.D.