Question

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

• A. $12$
• B. $-12$
• C. $6$
• D. none of these

Answer 1

Answer: (A)

$f^{\prime }(x)=6x^2+18x+\lambda$
Since $f(x)$ is a decreasing function of $x$ in $(-2,-1)$
$\therefore f^{\prime }(x)<0$
$\Rightarrow 6x^2+18x+\lambda <0$
The value of $\lambda$, should be such that the
$6x^2+18x+\lambda =0$ has roots $-2$ and $-1$
$\therefore \left(-2\right)\left(-1\right)=\frac{\lambda }{6}$
$\Rightarrow \lambda =12$