question_answer

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

A) \[\left[ 1,\text{ }3 \right]\]

B) \[[1\text{ },5]\]

C) \[\left[ 3,\text{ }5 \right]\]

D) none of these

Correct Answer: B

Solution :

[b] Clearly, form the graph, the range is \[[1,f(-1)]\equiv [1,5].\] If \[x<1,\,\,f(x)=-(x-1)-(x-2)=-2x+3.\] In this interval, \[f(x)\]is decreasing. If \[1\le x<2,f(x)=x-1-(x-2)=1.\] In this interval, \[f(x)\]is constant. If \[2\le x\le 3.\]\[f(x)=x-1+x-2=2x-3.\] In this interval. \[f(x)\]is increasing. \[\therefore \max f(x)=\] the greatest among \[f(-1)\] and \[f(3)=5\], \[\min f(x)=f(1)=1\] So, range= [1, 5]