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Question
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Options
monotonically increasing
monotonically decreasing
not monotonic
constant
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Solution
monotonically decreasing
\[ \text{If 1} < x < 2, \text { then } x > 1 \text { and }x < 2 . \]
\[ \Rightarrow x - 1 > 0 \text { and }x - 2 < 0\]
\[ \Rightarrow \left| x - 1 \right| = x - 1 \text { and }\left| x - 2 \right|=-\left( x - 2 \right)\]
\[\text { Now,}\]
\[f\left( x \right) = 2 \left| x - 1 \right| + 3 \left| x - 2 \right|\]
\[ = 2\left( x - 1 \right) - 3\left( x - 2 \right)\]
\[ = 2x - 2 - 3x + 6\]
\[ = - x + 4\]
\[f'\left( x \right) = - 1 < 0, \forall x \in \left( 1, 2 \right)\]
\[\text { So, }f\left( x \right) \text { is monotonically decreasing.}\]
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