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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 ?
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Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[\ f\left( x \right) = 5 x^\frac{3}{2} - 3 x^\frac{5}{2} , x > 0\]
\[f'\left( x \right) = \frac{15}{2} x^\frac{1}{2} - \frac{15}{2} x^\frac{3}{2} \]
\[ = \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right)\]
\[\text { Here }, 0, 1 \text { are the roots } .\]
\[\text { The possible intervals are }\left( - \infty , 0 \right),\left( 0, 1 \right)\text { and }\left( 1, \infty \right)...(1)\]
\[\text { For f(x) to be increasing, we must have}\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right) > 0\]
\[ \Rightarrow x \in \left( 0, 1 \right)\]
\[\text { So,f(x)is increasing on } \left( 0, 1 \right) . \]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \frac{15}{2} x^\frac{1}{2} \left( 1 - x \right) < 0\]
\[ \Rightarrow x \in \left( 1, \infty \right)\]
\[\text { So,f(x)is decreasing on }\left( 1, \infty \right).\]
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