The value of a for which twice the sum of the cubes of the roots of the equation a = x 2 - 3 x - 2 attains its minimum value is (where, a ∈ 0 , π )

Step-by-step solution

The given equation is x 2 - a x + 2 a - 3 = 0 Let α , β be the roots. Then, 2 α 3 + β 3 = 2 α + β 3 - 3 α β α + β = 2 a 3 - 3 2 a - 3 a = 2 a 3 - 6 a 2 + 9 a = 2 E (say) ∴ d E d a = 3 a 2 - 12 a + 9 = 3 a - 1 a - 3 Thus, E attains its minimum value at a = 3