Question

If f(x) = x³ + 4x² + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then ______.

Options

• λ = 4

• λ = 2

• λ = –1

• λ has no real value.

Answer 1

If f(x) = x³ + 4x² + λx + 1(λ ∈ R) is a monotonically decreasing function of x in the largest possible interval `(–2, (–2)/3)` then λ = 4.

Explanation:

f(x) = x³ + 4x² + λx + 1

Differentiating w.r.t. x, we get

f'(x) = 3x² + 8x + λ + 0 < 0 .......(i)

Also, `x∈(–2, (–2)/3)`

⇒ `(x + 2)(x + 2/3) < 0`

⇒ `x^2 + 2x + 2/3x + 4/3 < 0`

⇒ `x^2 + 8/3x + 4/3 < 0`

⇒ 3x² + 8x + 4 < 0  ...(ii)

Comparing equations (i) and (ii),

we get, λ = 4