In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is

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monotonically decreasing ext{If 1} < x < 2, ext { then } x > 1 ext { and }x < 2 . \Rightarrow x - 1 > 0 ext { and }x - 2 < 0 \Rightarrow \left| x - 1 ight| = x - 1 ext { and }\left| x - 2 ight|=-\left( x - 2 ight) ext { Now,} f\left( x ight) = 2 \left| x - 1 ight| + 3 \left| x - 2 ight| = 2\left( x - 1 ight) - 3\left( x - 2 ight) = 2x - 2 - 3x + 6 = - x + 4 f'\left( x ight) = - 1 < 0, \forall x \in \left( 1, 2 ight) ext { So, }f\left( x ight) ext { is monotonically decreasing.}

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