If $f\left(x\right)=\left|x\right|+|x-1|+|x-2|,$ then which one of the following is not correct?

We have,

$f\left(x\right)=\left|x\right|+|x-1|+|x-2|=\left\{\begin{array}{l}-3x+3,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x<0\\ -x+3,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}0\le x<1\\ x+1,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}1\le x<2\\ 3x-3,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}x\ge 2\end{array}\right.$

$\Rightarrow f^{\mathrm{\prime }}\left(x\right)=\left\{\begin{array}{r}-3,x<0\\ -1,0<x<1\\ 1,1<x<2\\ 3,x>2\end{array}\right.$

Clearly, $f\left(x\right)$ is not differentiable at $x=0, 1,2.$

It is evident from the definition that $f\left(x\right)$ attains a minimum at $x=1$ but it does not have a maximum or minimum at $x=0, 2.$