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Question
Find the interval in which the following function are increasing or decreasing f(x) = x3 − 6x2 + 9x + 15 ?
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Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = x^3 - 6 x^2 + 9x + 15\]
\[f'\left( x \right) = 3 x^2 - 12x + 9\]
\[ = 3 \left( x^2 - 4x + 3 \right)\]
\[ = 3 \left( x - 1 \right)\left( x - 3 \right)\]
\[\text { For f(x) to be increasing, we must have }\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow 3 \left( x - 1 \right)\left( x - 3 \right) > 0 \]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) > 0 \left[ \text { Since } 3 > 0, 3 \left( x - 1 \right)\left( x - 3 \right) > 0 \Rightarrow \left( x - 1 \right)\left( x - 3 \right) > 0 \right]\]
\[ \Rightarrow x < 1 \ or \ x > 3\]
\[ \Rightarrow x \in \left( - \infty , 1 \right) \cup \left( 3, \infty \right)\]
\[\text { So,f(x)is increasing on } x \in \left( - \infty , 1 \right) \cup \left( 3, \infty \right).\]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 3 \left( x - 1 \right)\left( x - 3 \right) < 0\]
\[ \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0 \left[ \text { Since } 3 > 0, 3 \left( x - 1 \right)\left( x - 3 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x - 3 \right) < 0 \right]\]
\[ \Rightarrow 1 < x < 3 \]
\[ \Rightarrow x \in \left( 1, 3 \right)\]
\[\text { So,f(x)is decreasing on x } \in \left( 1, 3 \right) .\]
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