Question

In the interval (1, 2), function f(x) = 2 | x − 1 | + 3 | x − 2 | is
(a) monotonically increasing
(b) monotonically decreasing
(c) not monotonic
(d) constant

Answer 1

(b) monotonically decreasing

$$\begin{gathered} \mathrm{If}1<x<2,\mathrm{then}x>1\text{and}x<2. \\ \Rightarrow x-1>0\text{and}x-2<0 \\ \Rightarrow \left|x-1\right|=x-1\text{and}\left|x-2\right|\text{=-}\left(x-2\right) \\ \text{Now,} \\ f\left(x\right)=2\left|x-1\right|+3\left|x-2\right| \\ =2\left(x-1\right)-3\left(x-2\right) \\ =2x-2-3x+6 \\ =-x+4 \\ f\text{'}\left(x\right)=-1<0,\forall x\in \left(1,2\right) \\ \text{So,}f\left(x\right)\text{is monotonically decreasing.} \end{gathered}$$