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

• A. $12$
• B. $-12$
• C. $6$
• D. $-6$

Answer 1

The correct option is A $12$

$f\left(x\right)=2x^3+9x^2+\lambda x+20$
Since f(x) is decreasing i.e. f'(x) < 0
$\Rightarrow 6x^2+18x+\lambda <0$
$\Rightarrow 6(x^2+3x+\frac{\lambda }{6})<0$
The value of $\lambda$ is such that the equation $x^2+3x+\frac{\lambda }{6}<0$ has roots - 2 and - 1
$\therefore$ Product of roots $=\frac{\lambda }{6}=(-2)(-1)$
$\Rightarrow \lambda =12$