Question

If $f\left(\right.x\left.\right)=x^{3}+4x^{2}+\lambda x+1$ is a monotonically decreasing function of $x$ in the largest possible interval $\left(\right.-2,-\frac{2}{3}\left.\right)$ , then

• A. $\lambda$ = 4
• B. $\lambda$ = 2
• C. $\lambda$ = -1
• D. $\lambda$ has no real value

Answer 1

Answer: (A)

Here, $f'(x) \leq 0$
$\Rightarrow 3( x )^{2}+8 x +\lambda \leq 0, \forall x \in\left(-2,-\frac{2}{3}\right)$
Then, situations for $f'(x)$ is as follow:

Given that $f(x)$ decreases in the largest possible interval $\left(-2,-\frac{2}{3}\right)$ then $f'(x)=0$ must have roots $-2$ and $-\frac{2}{3}$
$\Rightarrow$ Product of roots is $(-2)\left(-\frac{2}{3}\right)=\frac{\lambda}{3}$
$\Rightarrow \lambda=4$