Find the intervals in which the following functions are increasing or decreasing.

Theorem:- Let $f$ be a differentiable real function defined on an open interval $(a, b)$.

(i) If $f^{\prime}(x)>0$ for all $x \in(a, b)$, then $f(x)$ is increasing on $(a, b)$

(ii) If $f^{\prime}(x)<0$ for all $x \in(a, b)$, then $f(x)$ is decreasing on $(a, b)$

Algorithm:-

(i) Obtain the function and put it equal to $f(x)$

(ii) Find $f^{\prime}(x)$

(iii) Put $f^{\prime}(x)>0$ and solve this inequation.

For the value of $x$ obtained in (ii) $f(x)$ is increasing and for remaining points in its domain, it is decreasing.

Here we have,

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

$\Rightarrow \mathrm{f}(\mathrm{x})=\frac{\mathrm{d}}{\mathrm{dx}}\left(5 \mathrm{x}^{\frac{2}{2}}-3 \mathrm{x}^{\frac{5}{2}}\right)$

$\Rightarrow \mathrm{f}(\mathrm{x})=\frac{15}{2} \mathrm{x}^{\frac{1}{2}}-\frac{15}{2} \mathrm{x}^{\frac{3}{2}}$

$\Rightarrow \mathrm{f}(\mathrm{x})=\frac{15}{2} \mathrm{x}^{\frac{1}{2}}(1-\mathrm{x})$

For $f(x)$ lets find critical point, we must have

$\Rightarrow f^{\prime}(x)=0$

$\Rightarrow \frac{15}{2} x^{\frac{1}{2}}(1-x)=0$

$\Rightarrow x^{\frac{1}{2}}(1-x)=0$

$\Rightarrow x=0,1$

Since $x>0$, therefore only check the range on the positive side of the number line.

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

and $f^{\prime}(x)<0$ if $x>1$

Thus, $f(x)$ increases on $(0,1)$

and $f(x)$ is decreasing on interval $x \in(1, \infty)$