Thank you for reporting, we will resolve it shortly
Q.
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
Application of Derivatives
Solution:
$f'(x ) = 6x^2 + 18x + \lambda$
Since $f(x)$ is a decreasing function of $x$ in $(-2, -1)$
$\therefore f'(x) < 0$
$\Rightarrow 6x^2 + 18x + \lambda < 0$
The value of $\lambda$, should be such that the
$6x^2 + 18x + \lambda = 0$ has roots $- 2$ and $- 1$
$\therefore \left(-2\right)\left(-1\right) = \frac{\lambda}{6}$
$\Rightarrow \lambda = 12$