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Question
Find the interval in which the following function are increasing or decreasing f(x) = 2x3 − 12x2 + 18x + 15 ?
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Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = 2 x^3 - 12 x^2 + 18x + 15\]
\[f'\left( x \right) = 6 x^2 - 24x + 18\]
\[ = 6 \left( x^2 - 4x + 3 \right)\]
\[ = 6 \left( x - 1 \right)\left( x - 3 \right)\]
\[\text { For }f(x) \text { to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 6 \left( x - 1 \right)\left( x - 3 \right) > 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) > 0 \left[ \text { Since } 6 > 0, 6 \left( x - 1 \right)\left( x - 3 \right) > 0 \Rightarrow \left( x - 1 \right)\left( x - 3 \right) > 0 \right]\]
\[ \Rightarrow x < 1 \ or \ x > 3\]
\[ \Rightarrow x \in \left( - \infty , 1 \right) \cup \left( 3, \infty \right)\]
\[\text { So },f(x)\text { is increasing on }\left( - \infty , 1 \right) \cup \left( 3, \infty \right) . \]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6 \left( x - 1 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0 \left[ \text { Since} 6 > 0, 6 \left( x - 1 \right)\left( x - 3 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0 \right]\]
\[ \Rightarrow 1 < x < 3 \]
\[ \Rightarrow x \in \left( 1, 3 \right)\]
\[\text { So },f(x)\text { is decreasing on }\left( 1, 3 \right).\]
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