Question

Find the interval in which the following function are increasing or decreasing \[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\] ?

Answer 1

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

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

\[f\left( x \right) = \frac{3}{2} x^4 - 4 x^3 - 45 x^2 + 51\]

\[f'\left( x \right) = 6 x^3 - 12 x^2 - 90x\]

\[ = 6x\left( x^2 - 2x - 15 \right)\]

\[ = 6x\left( x - 5 \right)\left( x + 3 \right)\]

\[\text { Here, } x = - 3, x = 0 \text { and }x = 5 \text { are the critical points }.\]

\[\text { The possible intervals are }\left( - \infty , - 3 \right),\left( - 3, 0 \right),\left( 0, 5 \right)\text { and }\left( 5, \infty \right). .....(1)\]

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

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

\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) > 0 \left[\text {  Since,} 6 > 0, 6x\left( x - 5 \right)\left( x + 3 \right) > 0 \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) > 0 \right]\]

\[ \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) > 0\]

\[ \Rightarrow x \in \left( - 3, 0 \right) \cup \left( 5, \infty \right) \left[ \text { From eq.} (1) \right]\]

\[\text { So,f(x)is increasing on x } \in \left( - 3, 0 \right) \cup \left( 5, \infty \right) .\]

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

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

\[ \Rightarrow 6x\left( x - 5 \right)\left( x + 3 \right) < 0 \left[ \text { Since }6 > 0, 6x\left( x - 5 \right)\left( x + 3 \right) < 0 \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) < 0 \right]\]

\[ \Rightarrow x\left( x - 5 \right)\left( x + 3 \right) < 0\]

\[ \Rightarrow x \in \left( - \infty , - 3 \right) \cup \left( 0, 5 \right) \left[ \text { From eq.} (1) \right]\]

\[\text { So,f(x)is decreasing on x } \in \left( - \infty , - 3 \right) \cup \left( 0, 5 \right) .\]