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

$\text { Increasing interval }(0,1) \cup(2, \infty) \\ \text { Decreasing interval }(-\infty ,0) \cup(1,2)$

Given:

Here given that

$f(x)=x^{4}-4x^{3}+4x^{2}+15$

To find:

We have to find the increasing or decreasing interval for f(x).

Hint:

Put f ‘(x) = 0 to find critical points and then use increasing and decreasing property.

Solution:

We have,

$f(x)=x^{4}-4x^{3}+4x^{2}+15$

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

\begin{aligned} &f^{\prime}(x)=\frac{d}{d x}\left(x^{4}-4 x^{3}+4 x^{2}+15\right) \\ &\Rightarrow f^{\prime}(x)=4 x^{3}-12 x^{2}+8 x \end{aligned}

For critical points. we must have,

\begin{aligned} &f^{\prime}(x)=0 \\ &\Rightarrow 4 x^{3}-12 x^{2}+8 x=0 \\ &\Rightarrow 4 x\left(x^{2}-3 x+2\right)=0 \\ &\Rightarrow x\left(x^{2}-3 x+2\right)=0\{\therefore 4>0\} \\ &\Rightarrow x\left(x^{2}-2 x-x+2\right)=0 \\ &\Rightarrow x(x-2)(x-1)=0 \\ &\Rightarrow x=0, \quad x=1, \quad x=2 \end{aligned}

\begin{aligned} &\text { The possible intervals are }(-\infty, 0),(0,1),(1,2) \text { and }(2, \infty)\\ &\text { in intervals }(0,1) \text { and }(2, \infty), f^{\prime}(x) \end{aligned}

$\text { Clearly, } f^{\prime}(x)>0 \text { if } 0

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

$\text { Thus, } f(x) \text { is increasing on }(0,1) \cup(2, \infty) \text { and decreasing on }(-\infty, 0) \cup(1,2) \text { . }$