If f ( x ) = x 3 + 4 x 2 + λx + 1 is a monotonically decreasing function of x in the largest possible interval ( - 2 , - 2 3 ) , then

Question

If f ( x ) = x 3 + 4 x 2 + λx + 1 is a monotonically decreasing function of x in the largest possible interval ( - 2 , - 2 3 ) , then, λ = 4, λ = 2, λ = -1, λ has no real value

MARKS App · Accepted Answer

Here, f ' ( x ) ≤ 0 ⇒ 3 x 2 + 8 x + λ ≤ 0 , ∀ x ∈ - 2 , - 2 3 Then, situations for f ' ( x ) is as follow : Given that f ( x ) decreases in the largest possible interval - 2 , - 2 3 then f ' ( x ) = 0 must have roots - 2 and - 2 3 ⇒ Product of roots is ( - 2 ) - 2 3 = λ 3 ⇒ λ = 4

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