Question

Find range of $f\left(x\right)=|x+1|+|2x+1|+|2x-1|+|x-1|$

Answer 1

If number of terms are even than there will be two middle terms as $|2x+1|$ and $|2x–1|$

$(2x+1)=0\Rightarrow x=\frac{-1}{2}$

$(2x-1)=0\Rightarrow x=\frac{1}{2}$

Let $m\in [\frac{-1}{2},\frac{1}{2}]$

$m$ can be any value between $[\frac{-1}{2},\frac{1}{2}]$ then range $\in [f\left(m\right),\infty )$

generally we choose value of $m$ at which calculations are easy like we can take $m=0$.

$f\left(0\right)=|0+1|+|0+1|+|2.0-1|+|0-1|=4$

$\therefore$ Range$\in [4,\infty )$