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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

Options

  1. Aλ = 4
  2. Bλ = 2
  3. Cλ = -1
  4. Dλ has no real value

Correct answer

A. λ = 4

Step-by-step solution

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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