Question

The function f(x) = |x| - |x-1| is monotonically increasing when .

• A. x < 0
• B. 0 < x < 1
• C. x > 1
• D. x < 1

Answer 1

The correct option is B 0 < x < 1

Given the function f(x) = |x| - |x-1|.
$f\left(x\right)=(\begin{array}{c}-x-\left(-(x-1)\right],\mathrm{if}x\le 0\\ x-\left(-(x-1)\right],\mathrm{if}0<x<1\\ x-\left(x-1\right),\mathrm{if}x\ge 1\end{array}\Rightarrow f(x)=(\begin{array}{c}-x+x-1,\mathrm{if}x\le 0\\ x+x-1,\mathrm{if}0<x<1\\ x-x+1,\mathrm{if}x\ge 1\end{array}\Rightarrow f\left(x\right)=(\begin{array}{c}-1,\mathrm{if}x\le 0\\ 2x-1,\mathrm{if}0<x<1\\ 1,\mathrm{if}x\ge 1\end{array}f\text{'}(x)=(\begin{array}{c}0,\mathrm{if}x\le 0\\ 2,\mathrm{if}0<x<1\\ 0,\mathrm{if}x\ge 1\end{array}\because f\text{'}\left(x\right)>0\mathrm{for}0<x<1.\therefore f\left(x\right)\mathrm{is}\mathrm{monotonically}\mathrm{increasing}\mathrm{for}0<x<1.$