Question

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

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

Answer 1

Answer: (A)

$\lambda =4$

$f\left(x\right)$ is monotonically decreasing so $f^{\prime }\left(x\right)<0$
$f^{\prime }\left(x\right)=3x^2+8x+\lambda <0$ in interval $(-2,-2/3)$
Substitute $f^{\prime }(-2)$ will get $\lambda <4$
But in the interval $\lambda =4$