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

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

Given:

Here given that

$f(x)=5 x^{\frac{3}{2}}-3 x^{\frac{5}{2}}, x>0$

To find:

We have to find the increasing and decreasing intervals.

Hint:

We will find the critical points and then use increasing and decreasing property.

Solution:

We have,

$f(x)=5 x^{\frac{3}{2}}-3 x^{\frac{5}{2}}$

Differentiating w.r.t. x we get,

\begin{aligned} &f^{\prime}(x)=\frac{d}{d x}\left(5 x^{\frac{3}{2}}-3 x^{\frac{5}{2}}\right) \\ &\Rightarrow f^{\prime}(x)=5\left(\frac{3}{2} x^{\frac{1}{2}}\right)-3\left(\frac{5}{2} x^{\frac{3}{2}}\right) \\ &\Rightarrow f^{\prime}(x)=\frac{15}{2} x^{\frac{1}{2}}-\frac{15}{2} x^{\frac{3}{2}} \\ &\Rightarrow f^{\prime}(x)=\frac{15}{2}\left(x^{\frac{1}{2}}-x^{\frac{3}{2}}\right) \end{aligned}

For f(x), we have to find critical points.

We must have,

\begin{aligned} &f^{\prime}(x)=0 \\ &\Rightarrow \frac{15}{2}\left(x^{\frac{1}{2}}-x^{\frac{3}{2}}\right)=0,\left\{\therefore \frac{15}{2}=0\right\} \\ &\Rightarrow x^{\frac{1}{2}}-x^{\frac{3}{2}}=0 \\ &\Rightarrow x^{\frac{1}{2}}\left(1-x^{\frac{3}{2}}\right)=0 \end{aligned}

\begin{aligned} &\Rightarrow x=0 \text { and } x=1 \\ &\text { Clearly, } f^{\prime}(x)>0, \text { if } 01 \text { or } x \in(1, \infty) \end{aligned}

$\text { Thus, } f(x) \text { is increasing on the interval }(0,1) \text { and decreasing on the interval }(1, \infty)$