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

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

$f(x)=2 x^{3}+9 x^{2}+12 x+20$

Solution:

Given:- Function $f(x)=2 x^{3}+9 x^{2}+12 x+20$

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

(i) If $f^{\prime}(x)>0$ for all $x \in(a, b)$, 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.

Here we have,

$f(x)=2 x^{3}+9 x^{2}+12 x+20$

$\Rightarrow f(x)=\frac{d}{d x}\left(2 x^{3}+9 x^{2}+12 x+20\right)$

$\Rightarrow f^{\prime}(x)=6 x^{2}+18 x+12$

For $f(x)$ lets find critical point, we must have

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

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

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

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

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

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

$\Rightarrow x=-1,-2$

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

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

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

and $f(x)$ is decreasing on interval $(-\infty,-2) \cup(-2, \infty)$