Question

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
(a) increasing
(b) decreasing
(c) constant
(d) none of these

Answer 1

(b) decreasing

$$\begin{gathered} \text{Given:}f\left(x\right)=2\left|x-1\right|+3\left|x-2\right| \\ \mathrm{If}1\mathit{<}x\mathit{<}\mathit{2}\mathit{,}\mathit{}\mathrm{t}\text{hen}\left|x-1\right|\mathit{=}x-1. \\ \Rightarrow \left|x\mathit{-}\mathit{2}\right|\mathit{=}\mathit{-}\left(x\mathit{-}\mathit{2}\right) \\ \text{Now,} \\ f\left(x\right)=2\left|x\mathit{-}1\right|+3\left|x\mathit{-}2\right| \\ =2\left(x\mathit{-}1\right)+3\left(\mathit{-}x+2\right) \\ =2x\mathit{-}2\mathit{-}3x+6 \\ =\mathit{-}x+4 \\ f\mathit{\text{'}}\left(x\right)\mathit{=}\mathit{-}\mathit{1}\mathit{<}\mathit{0} \\ \text{So,}f\left(x\right)\text{is decreasing when}1<x<2. \end{gathered}$$