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

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

Given:

Here given that

$f(x)=x^{4}-4x$

To find:

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

Hint:

We will find critical points.

Solution:

We have,

$f(x)=x^{4}-4x$

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

\begin{aligned} &f^{\prime}(x)=\frac{d}{d x}\left(x^{4}-4 x\right) \\ &\Rightarrow f^{\prime}(x)=4 x^{3}-4 \end{aligned}

For critical points we must have,

\begin{aligned} &f^{\prime}(x)=0 \\ &\Rightarrow 4\left(x^{3}-1\right)=0 \\ &\Rightarrow x^{3}-1=0\{\therefore 4>0\} \\ &\Rightarrow x^{3}=1 \end{aligned}

Taking cube root on both sides.

\begin{aligned} &\Rightarrow \sqrt[3]{x}=\sqrt[3]{1} \\ &\Rightarrow x=1 \\ &\text { Clearly, } f^{\prime}(x)>0 \text { if } x>1 \text { or } x \in(1, \infty) \text { and } f^{\prime}(x)<0 \text { if } x<1 \text { or } x \in(-\infty, 1) . \end{aligned}

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