Let λ * be the largest value of λ for which the function f λ x = 4 λ x 3 - 36 λ x 2 + 36 x + 48 is increasing for all x ∈ ℝ . Then f λ * 1 + f λ , * - 1 is equal to:

Step-by-step solution

Given, f λ x = 4 λ x 3 - 36 λ x 2 + 36 x + 48 Differentiating w.r.t x both side we get, f λ ' x = 12 λ x 2 - 72 λ x + 36 f λ x ' = 12 λ x 2 - 6 λ x + 3 ≥ 0 ∴   λ > 0 and D ≤ 0 36 λ 2 - 4 × λ × 3 ≤ 0 9 λ 2 - 3 λ ≤ 0 3 λ 3 λ - 1 ≤ 0 λ ∈ 0 , 1 3 So,   λ largest   = 1 3 f x = 4 3 x 3 - 12 x 2 + 36 x + 48 So,   f 1 + f - 1 = 72