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Question
Function f(x) = 2x3 − 9x2 + 12x + 29 is monotonically decreasing when
Options
x < 2
x > 2
x > 3
1 < x < 2
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Solution
1 < x < 2
\[f\left( x \right) = 2 x^3 - 9 x^2 + 12x + 29\]
\[f'\left( x \right) = 6 x^2 - 18x + 12\]
\[ = 6 \left( x^2 - 3x + 2 \right)\]
\[ = 6\left( x - 1 \right)\left( x - 2 \right)\]
\[\text { For f(x) to be decreasing, we must have }\]
\[f'\left( x \right) < 0\]
\[ \Rightarrow 6\left( x - 1 \right)\left( x - 2 \right) < 0 \]
\[ \Rightarrow \left( x - 1 \right)\left( x - 2 \right) < 0 \left[ \text { Since }6 > 0, 6\left( x - 1 \right)\left( x - 2 \right) < 0 \Rightarrow \left( x - 1 \right)\left( x - 2 \right) < 0 \right]\]
\[ \Rightarrow 1 < x < 2\]
\[\text { So,f(x) is decreasing for }1 < x < 2 .\]
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