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

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

Given:

Here given that

$f(x)=2x^{3}-9x^{2}+12x-5$

To find:

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

Hint:

Put f ‘(x) = 0 and solve this equation to find critical points of f(x).

Solution:

We have,

$f(x)=2x^{3}-9x^{2}+12x-5$

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

\begin{aligned} &f^{\prime}(x)=\frac{d}{d x}\left(2 x^{3}-9 x^{2}+12 x-5\right) \\ &f \Rightarrow^{\prime}(x)=6 x^{2}-18 x+12 \end{aligned}

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

\begin{aligned} &f^{\prime}(x)=0\\ &\Rightarrow 6 x^{2}-18 x+12=0\\ &\Rightarrow 6\left(x^{2}-3 x+2\right)=0\\ &\Rightarrow x^{2}-3 x+2=0\{\therefore 6>0\}\\ &\Rightarrow x^{2}-2 x-x+2=0\\ &\Rightarrow(x-1)(x-2)=0\\ &\Rightarrow x-1=0 \text { and } x-2=0\\ &\Rightarrow x=1 \text { and } x=2\\ &\text { Clearly, } f^{\prime}(x)>0 \text { if } x<1 \text { and } x>2 \text { or } x \in(-\infty, 1) \text { and } x \in(2, \infty) \text { and } f^{\prime}(x)<0 \text { if } \end{aligned}

\begin{aligned} &1