The least value of the product x y z (such that x , y and z are positive real numbers) for which the determinant x 1 1 1 y 1 1 1 z is non-negative is

Step-by-step solution

Since A . M   ≥ G . M x + y + z 3   ≥ x y z 1 3 x + y + z   ≥ 3 x y z 1 3 Now, x 1 1   1 y 1   1 1 z = x y z - x + y + z + 2 Now determinant is non-negative ⇒ x y z - x + y + z + 2 ≥ 0 ⇒ x y z - 3 x y z 1 3 + 2   ≥ 0 Let x y z = t 3 So, t 3 - 3 t + 2   ≥ 0 ⇒ t + 2 ( t 2 - 2 t + 1 ) ≥ 0 ⇒ t + 2 ( t - 1 ) 2 ≥ 0 ⇒ t ≥ - 2 ⇒ t 3 ≥ - 8 ⇒ Least value of x y z =   - 8