Use HM ≤ AM and find maximum value of xyx+y + yzy+z + xzx+z
From AM≥GM≥HM
We know that AM ≥ HM and GM ≥ HM
(i) ⇒ x+y2 ≥ 2xyx+y ⇒ xyx+y ≤ 14(x+y) --------(1)
y+z2 ≥ 2yzy+z ⇒ yzy+z ≤ 14(y + z) --------(2)
Similarly xzx+z ≤ 14(x + z) --------------------(3)
Adding (1), (2) and (3),
xyx+y + yzy+z + xzx+z ≤ 14(2x + 2y + 2z)
≤ 12(x + y + z)