If λ ∈ R is such that the sum of the cubes of the roots of the equation x 2 + 2 - λ x + 10 - λ = 0 is minimum, then the magnitude of the difference of the roots of this equation is :

Step-by-step solution

x 2 + 2 - λ x + 10 - λ = 0 ⇒ α + β = λ - 2 &   α β = 10 - λ   . . . . . . . . . . i Let roots are α   a n d   β . ⇒   α 3 + β 3 = α + β 3 - 3 α β ( α + β ) = λ - 2 3 - 3 10 - λ ( λ - 2 ) =   λ 3 - 6 λ 2 + 12 λ - 8 - 3 ( 10 λ - λ 2 - 20 + 2 λ ) =   λ 3 - 3 λ 2 - 24 λ + 52 d z d λ = 3 λ 2 - 6 λ - 24 = 3 &