Find the intervals in which the following functions are increasing or decreasing.

Solution:

Given:- Function $f(x)=-2 x^{3}-9 x^{2}-12 x+1$

Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$.

(i) If $f^{\prime}(x)>0$ for $a l l x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$

(ii) If $f^{\prime}(x)<0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$

Algorithm:-

(i) Obtain the function and put it equal to $f(x)$

(ii) Find $f^{\prime}(x)$

(iii) Put $f^{\prime}(x)>0$ and solve this inequation.

For the value of $x$ obtained in (ii) $f(x)$ is increasing and for remaining points in its domain it is decreasing.

$\Rightarrow f^{\prime}(x)=0$

$\Rightarrow-6 x^{2}-18 x-12=0$

$\Rightarrow 6 x^{2}+18 x+12=0$

$\Rightarrow 6\left(x^{2}+3 x+2\right)=0$

$\Rightarrow 6\left(x^{2}+2 x+x+2\right)=0$

$\Rightarrow x^{2}+2 x+x+2=0$

$\Rightarrow(x+2)(x+1)=0$

$\Rightarrow x=-1,-2$

clearly, $f^{\prime}(x)>0$ if $x<-2$ and $x>-1$

and $f^{\prime}(x)<0$ if $-2

Thus, $f(x)$ increases on $(-\infty,-2) \cup(-1, \infty)$

and $f(x)$ is decreasing on interval $x \in(-2,-1)$