The number of different possible values for the sum x + y + z , where x ,   y ,   z are real number such that x 4 + 4 y 4 + 16 z 4 + 64 = 32 x y z is

Step-by-step solution

We have, x 4 + 4 y 4 + 16 z 4 + 64 = 32 x y z We know A M ≥ G M ⇒ x 4 + 4 y 4 + 16 z 4 + 64 4 ≥ x 4 × 4 y 4 × 16 z 4 × 64 1 / 4 ⇒ x 4 + 4 y 4 + 16 z 4 + 64 ≥ 32 x y z But given x 4 + 4 y 4 + 16 z 4 + 64 = 32 x y z ∴ x 4 = 4 y 4 = 16 z 4 = 64 ⇒ x = ± 2 2 , y = ± 2 , z = ± 2 For x ,   y ,   z For x 4 + 4 y 4 + 16 z 4 + 64 = 32 x y z Either each of x ,   y ,   z is positive → 1 case or two of x ,   y ,   z are n