Question

$f\left(x\right)=|x-1|+|x-3|$, then find $f^{\prime }\left(x\right)$

Answer 1

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

$=\{\begin{array}{cc}-(x-1)-(x-3),& x<1\\ 2,& x=1\\ (x-1)-(x-3),& 1<x<3\\ 2,& x=3\\ (x-1)+(x-3),& x>3\end{array}$

$=\{\begin{array}{cc}-2x+4,& x<1\\ 2,& x=1\\ 2,& 1<x<3\\ 2,& x=3\\ 2x-4,& x>3\end{array}$

$\therefore f^{\prime }\left(x\right)=\{\begin{array}{cc}-2,& x<1\\ 0,& 1\le x\le 3\\ 2,& x>3\end{array}$.