Question

Find the interval in which the following function are increasing or decreasing  f(x) =  \[5 x^\frac{3}{2} - 3 x^\frac{5}{2}\]  x > 0 ?

Answer 1

\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]

\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]

\[\ f\left( x \right) = 5 x^\frac{3}{2} - 3 x^\frac{5}{2} , x > 0\]

\[f'\left( x \right) = \frac{15}{2} x^\frac{1}{2} - \frac{15}{2} x^\frac{3}{2} \]

\[ = \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right)\]

\[\text { Here }, 0, 1 \text { are the roots } .\]

\[\text { The possible intervals are }\left( - \infty , 0 \right),\left( 0, 1 \right)\text { and }\left( 1, \infty \right)...(1)\]

\[\text { For f(x) to be increasing, we must have}\]

\[f'\left( x \right) > 0\]

\[ \Rightarrow \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right) > 0\]

\[ \Rightarrow x \in \left( 0, 1 \right)\]

\[\text { So,f(x)is increasing on } \left( 0, 1 \right) . \]

\[\text { For f(x) to be decreasing, we must have }\]

\[f'\left( x \right) < 0\]

\[ \Rightarrow \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right) < 0\]

\[ \Rightarrow x \in \left( 1, \infty \right)\]

\[\text { So,f(x)is decreasing on }\left( 1, \infty \right).\]