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

$\text { Increasing interval }(0, \infty)\\ \text { Decreasing interval }(-\infty,0)$

Given:

Here given that

$f(x)=x^{8}+6x^{z}$

To find:

We have to find the increasing and decreasing intervals.

Hint:

First find the critical points by using f ‘(x) and then apply increasing and decreasing property.

Solution:

We have,

$f(x)=x^{8}+6x^{z}$

Differentiating w.r.t. x we get,

\begin{aligned} &f^{\prime}(x)=\frac{d}{d x}\left(x^{8}+6 x^{2}\right) \\ &=8 x^{7}+12 x \end{aligned}

For critical points. We must have,

\begin{aligned} &f^{\prime}(x)=0 \\ &\Rightarrow 8 x^{7}+12 x=0 \\ &\Rightarrow 4 x\left(2 x^{6}+3\right)=0,\{\therefore 4=0\} \\ &\Rightarrow x\left(2 x^{6}+3\right)=0 \\ &\Rightarrow x=0,\left\{\therefore 2 x^{6}+3>0\right\} \\ &\text { Clearly, } f^{\prime}(x)>0, \text { if } x>0 \text { or } x \in(0, \infty) \\ &\text { and } f^{\prime}(x)<0 \text { if } x<0 \text { or } x \in(-\infty, 0) \end{aligned}

$\text { Thus, } f(x) \text { is increasing on the interval }(0, \infty) \text { and decreasing on the interval }(-\infty, 0)$