If - 3 , - 1 is the largest interval in which the function f x = x 3 + 6 x 2 + a x + 2 is decreasing, then a is equal to (where, . denotes the greatest integer function)

Step-by-step solution

f ′ x = 3 x 2 + 12 x + a if f x is decreasing in − 3 , − 1 , f ' x < 0 ,   ∀ x ∈ − 3 , − 1 ∴ f ′ − 3 = f ′ − 1 = 0 ⇒ a 3 = - 3 - 1 ⇒ a = 9 (using product of roots)