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

$\text { Increasing interval }(-\infty, 2) \cup(3, \infty) \\ \text { Decreasing interval (2,3) }$

Given:

Here given that

$f(x)=2x^{3}-15x^{2}+36x+1$

To find:

We have to find the intervals in which function is increasing and decreasing.

Hint:

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

Solution:

We have,

$f(x)=2x^{3}-15x^{2}+36x+1$

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

\begin{aligned} &f^{\prime}(x)=\frac{d}{d x}\left(2 x^{3}-15 x^{2}+36 x+1\right) \\ &f \Rightarrow f^{\prime}(x)=6 x^{2}-30 x+36 \end{aligned}

Now for critical points of f(x), we must have,

\begin{aligned} &f^{\prime}(x)=0 \\ &\Rightarrow 6 x^{2}-30 x+36=0 \\ &\Rightarrow 6\left(x^{2}-5 x+6\right)=0 \\ &\Rightarrow x^{2}-5 x+6=0\{\therefore 6>0\} \\ &\Rightarrow x^{2}-3 x-2 x+6=0 \\ &\Rightarrow(x-3)(x-2)=0 \\ &\Rightarrow x-3=0 \text { and } x-2=0 \\ &\Rightarrow x=3 \text { and } x=2 \end{aligned}

\begin{aligned} &\text { Clearly, } f^{\prime}(x)>0 \text { if } x<2 \text { and } x>3 \text { or } x \in(-\infty, 2) \text { and } x \in(3, \infty) \text { and } f^{\prime}(x)<0 \text { if }\\ &2

$\text { So, } f(x) \text { is increasing on the interval } (-\infty, 2)\: \cup\: (3, \infty) \text { and } f(x) \text { is decreasing on interval }(2,3).$