$F(x) = |x-1|+|x^2-2x|$ domain $[0,2]$ max and min?

Question

When $0<x<1 $ the critical point is $x=0.5$ and $y = 5/4$ When $1<x<2 $ the critical point is $x=1.5$ and $y = 5/4$

$F(0) = 1$ and $F(2) = 1$ Do I have to include the boundaries in the domain? If it written $[0,2]$ do I have to test $0$ and $2$? Is the max $5/4$ and the min $1$?