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Question
Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?
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Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\]
\[f'\left( x \right) = 6 x^3 - 12 x^2 - 90x\]
\[ = 6x\left( x^2 - 2x - 15 \right)\]
\[ = 6x\left( x - 5 \right)\left( x + 3 \right)\]
\[\text { Here, } x = - 3, x = 0 \text { and }x = 5 \text { are the critical points }.\]
\[\text { The possible intervals are }\left( - \infty , - 3 \right),\left( - 3, 0 \right),\left( 0, 5 \right)\text { and }\left( 5, \infty \right). .....(1)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) > 0 \left[\text { Since,} 6 > 0, 6x\left( x - 5 \right)\left( x + 3 \right) > 0 \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) > 0 \right]\]
\[ \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) > 0\]
\[ \Rightarrow x \in \left( - 3, 0 \right) \cup \left( 5, \infty \right) \left[ \text { From eq.} (1) \right]\]
\[\text { So,f(x)is increasing on x } \in \left( - 3, 0 \right) \cup \left( 5, \infty \right) .\]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) < 0 \left[ \text { Since }6 > 0, 6x\left( x - 5 \right)\left( x + 3 \right) < 0 \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) < 0 \right]\]
\[ \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) < 0\]
\[ \Rightarrow x \in \left( - \infty , - 3 \right) \cup \left( 0, 5 \right) \left[ \text { From eq.} (1) \right]\]
\[\text { So,f(x)is decreasing on x } \in \left( - \infty , - 3 \right) \cup \left( 0, 5 \right) .\]
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