Question

The range of the function $f\left(x\right)=|x-1|+|x-2|,-1\le x\le 3$, is

• A. $[2,5]$
• B. $[1,5]$
• C. $[3,5]$
• D. None of these

Answer 1

The correct option is B

$[1,5]$

Find the range of the given function

Given, $f\left(x\right)=\left|x-1\right|+\left|x-2\right|$

For, $-1\le x\le 1$

$f\left(x\right)=-(x-1)-(x-2)=-2x+3$

For, $1\le x\le 2$

$f\left(x\right)=(x-1)-(x-2)$

$=1$

For, $2\le x\le 3$

$f\left(x\right)=(x-1)+(x-2)=2x-3$

$\therefore f\left(x\right)$ has the maximum value at, $x=-1$

$\Rightarrow f(-1)=-2(-1)+3=5$

$\therefore f\left(x\right)$has the minimum value at, $x=1$ and $2$

$$\begin{gathered} \Rightarrow ƒ\left(1\right)=1 \\ \Rightarrow ƒ\left(2\right)=2\left(2\right)-3=1 \end{gathered}$$

$f\left(1\right)=f\left(2\right)=1$

The range of the function is $\left[1,5\right]$

Hence, the correct answer is $B$, $\left[1,5\right]$.