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Question
In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
Options
increasing
decreasing
constant
none of these
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Solution
decreasing
\[\text { Given }: f\left( x \right) = 2\left| x - 1 \right| + 3\left| x - 2 \right|\]
\[\text { If 1 < x < 2, then }\left| x - 1 \right| = x - 1 . \]
\[ \Rightarrow \left| x - 2 \right| = - \left( x - 2 \right)\]
\[\text { Now,}\]
\[f\left( x \right) = 2\left| x - 1 \right| + 3\left| x - 2 \right|\]
\[ = 2 \left( x - 1 \right) + 3 \left( - x + 2 \right)\]
\[ = 2x - 2 - 3x + 6\]
\[ = - x + 4\]
\[f'\left( x \right) = - 1 < 0\]
\[\text { So,}f\left( x \right) \text { is decreasing when 1 < x < 2 } .\]
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