Find the intervals in which function f (x)=5 x^(3/2)-3 x^(5/2) is (i) increasing (ii) decreasing.

\[\mathrm{f}(x)=5 x^{3 / 2}-3 x^{5 / 2} \Rightarrow \mathrm{f}^{\prime}(x)=\frac{15}{2} \sqrt{x}(1-x) \]

For increasing / decreasing,put $\displaystyle \boldsymbol{\mathrm{f}}^{\prime}(\boldsymbol{x})=\mathbf{0}$

\[\Rightarrow x=0,1 \]

When $\displaystyle x \in[0,1], \mathrm{f}^{\prime}(x) \geq 0$. So, $\displaystyle \mathrm{f}$ is increasing when $\displaystyle x \in[0,1]$

(The intervals( $\displaystyle 0,1 ),[0,1) \operatorname{or}(0,1]$ can alsobeconsidered.)

When $\displaystyle x \in[1, \infty), \mathrm{f}^{\prime}(x) \leq 0$.So, $\displaystyle \mathrm{f}$ is decreasing when $\displaystyle x \in[1, \infty)$

(The interval( $\displaystyle 1, \infty$ )can also beconsidered.)