Question

If $f\left(x\right)=2x^3+9x^2+\lambda x+20$ is a decreasing function of $x$ in the largest possible interval $(-2,-1)$ then $\lambda$ is equal to

Answer 1

Since $f\left(x\right)$ is decreasing in the interval $(-2,-1)$ , therefore,

$f^{\prime }\left(x\right)<0\Rightarrow 6x^2+18x+\lambda <0$.

The value of $\lambda$ should be such that the equation

$6x^2+18x+\lambda =0$ has roots $-2$ and $-1$.

Therefore, $(-2)(-1)=\frac{\lambda }{6}\Rightarrow \lambda =12$.