If the equation x 3 - 6 x 2 + 9 x + λ = 0 has exactly one root in 1,3 , then λ belongs to the interval

Step-by-step solution

Let, f x = x 3 - 6 x 2 + 9 x + λ ⇒ f ′ x = 3 x 2 − 12 x + 9 = 3 x 2 − 4 x + 3 = 3 x - 1 x - 3 f ' x < 0 ,   ∀ x ∈ 1 , 3 f 1 = 4 + λ ,   f 3 = λ For f x = 0 to have exactly one root in 1,3 , the values of f 1 and f 3 should have opposite signs ⇒ f 1 ⋅ f 3 < 0 ⇒ λ + 4 λ < 0 ⇒ λ ∈ - 4,0