Let α , β and γ be the roots of f x = 0 , where f x = x 3 + x 2 - 5 x - 1 . Then the value of α + β + γ is equal to (where ⋅ denotes the greatest integer function)

Step-by-step solution

f x = x 3 + x 2 - 5 x - 1 f ′ x = 3 x 2 + 2 x − 5 = x − 1 3 x + 5 x = - 5 3 → point of local maxima, x = 1 → point of local minima, Increasing function in - ∞ , - 5 3 ∪ 1 , ∞ Decreasing function in - 5 3 , 1 Also, f 0 = - 1 , f 1 = - 4 , f 2 = 1 , f - 1 = 5 f - 2 = - 8 + 4 + 10 - 1 = 5 f - 3 = - 27 + 9 + 15 - 1 = - 4 ⇒ α ∈ - 3 , - 2 ; β ∈ - 1,0 ; γ ∈ 1,2 ⇒ α = - 3 , β = - 1 , γ = 1 ⇒ α + β + γ = - 3