#### Provide solution for RD Sharma maths class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xiv

$\text { Increasing interval (-2,-1) } \\ \text { Decreasing interval }(-\infty,-2) \cup(-1, \infty)$

Given:

Here given that

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

To find:

We have to find the increasing and decreasing intervals of f(x).

Hint:

Put f ‘(x) = 0 to find the critical points.

Solution:

We have,

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

On differentiating we get,

\begin{aligned} &f^{\prime}(x)=\frac{d}{d x}\left(-2 x^{3}-9 x^{2}-12 x+1\right) \\ &\Rightarrow f^{\prime}(x)=-6 x^{2}-18 x-12 \end{aligned}

For f(x), we have to find critical points.

We must have,

\begin{aligned} &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 x^{2}+2 x+x+2=0 \\ &\Rightarrow x(x+2)+1(x+2)=0 \\ &\Rightarrow(x+2)(x+1)=0 \\ &\Rightarrow x+2=0 \text { and } x+1=0 \\ &\Rightarrow x=-2 \text { and } x=-1 \end{aligned}

\begin{aligned} &\text { Clearly, } f^{\prime(x)}<0, \text { if } x<-1 \text { and } x<-2 \text { or } x \in(-\infty,-2) \text { and } x \in(-1, \infty) \\ &\text { and } f^{\prime}(x)>0 \text { if }-2

$\text { Thus, } f(x) \text { is increasing on the interval }(-2,-1) \text { and decreasing on the interval }(-\infty,-2)\:\: \cup\:\: (-1,\infty )$