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Question
Find the interval in which the following function are increasing or decreasing f(x) = (x − 1) (x − 2)2 ?
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Solution
\[\text { When } \left( x - a \right)\left( x - b \right)>0 \text { with }a < b, x < a \text { or }x>b.\]
\[\text { When } \left( x - a \right)\left( x - b \right)<0 \text { with } a < b, a < x < b .\]
\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right)^2 \]
\[ = \left( x - 1 \right)\left( x^2 - 4x + 4 \right)\]
\[ = x^3 - 5 x^2 + 8x - 4\]
\[f'\left( x \right) = 3 x^2 - 10x + 8\]
\[ = 3 x^2 - 6x - 4x + 8\]
\[ = \left( x - 2 \right)\left( 3x - 4 \right)\]
\[\]
\[ \text{For f(x) to be increasing, we must have}\]
\[f'\left( x \right) > 0\]
\[ \Rightarrow \left( x - 2 \right)\left( 3x - 4 \right) > 0\]
\[ \Rightarrow x < \frac{4}{3} or x > 2\]
\[ \Rightarrow x \in \left( - \infty , - \frac{4}{3} \right) \cup \left( 2, \infty \right)\]
\[\text{So,f(x)is increasing on x }\in \left( - \infty , \frac{4}{3} \right) \cup \left( 2, \infty \right).\]
\[\]
\[\]
\[\text { For }f(x) \text { to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow \left( x - 2 \right)\left( 3x - 4 \right) < 0\]
\[ \Rightarrow \frac{4}{3} < x < 2 \]
\[ \Rightarrow x \in \left( \frac{4}{3}, 2 \right)\]
\[\text{So,f(x)is decreasing on x }\in \left( \frac{4}{3}, 2 \right) .\]
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