Let x , y , z be positive real numbers such that x + y + z = 12 and x 3 y 4 z 5 = 0 .1 600 3 . Then x 3 + y 3 + z 3 is equal to

Step-by-step solution

x + y + z = 12 A . M . ≥ G . M . 3 x 3 + 4 y 4 + 5 z 5 12   ≥   x 3 3   y 4 4   z 5 5 12   1 ≥ 1 ⇒ A . M . = G . M . ⇒ All the numbers are equal. ∴ x 3 = y 4 = z 5 = k x = 3 k ;   y = 4 k ;   z = 5 k x + y + z = 12 ⇒ 3 k + 4 k + 5 k = 12 ⇒ k = 1 ∴ x = 3 ;   y = 4;   z = 5 Therefore, x 3 + y 3 + z 3 = 216 .