#### Please solve RD Sharma class 12 chapter Increasing and Decreasing Functions exercise 16.2 question 1 subquestion xvii maths textbook solution

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

Given:

Here given that

$f(x)=2x^{3}-24x+7$

To find:

We have to find the intervals in which f(x) is increasing or decreasing.

Hint:

First we will find critical points and then use increasing and decreasing property.

Solution:

We have,

$f(x)=2x^{3}-24x+7$

Differentiating w.r.t. x, we get,

\begin{aligned} &f^{\prime}(x)=\frac{d}{d x}\left(2 x^{3}-24 x+7\right) \\ &\Rightarrow f^{\prime}(x)=6 x^{2}-24 \end{aligned}

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

We must have,

\begin{aligned} &f^{\prime}(x)=0\\ &\Rightarrow 6 x^{2}-24=0\\ &\Rightarrow 6 x^{2}=24\\ &\Rightarrow x^{2}=\frac{24}{6}\\ &\Rightarrow x^{2}=4\\ &\Rightarrow x=\pm 2\\ &\Rightarrow x=+2,-2\\ &\text { Clearly, } f^{\prime}(x)>0 \text { if } x>2 \text { and } x<-2 \text { or } x \in(-\infty, 2) \text { and } x \in(-2, \infty) \text { and } f^{\prime}(x)<0 \text { if }\\ &-2

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