Let f(x)=|x^2-1|+|x-2| for x∈mathbb R.

The kinks come from $x=\pm1$ (zeros of $x^2-1$, simple zeros) and $x=2$, so there are exactly three points of non-differentiability; (A) is true. On $[-1,1]$, $f=3-x-x^2$ (decreasing to $1$ at $x=1$); on $[1,2]$, $f=x^2-x+1\ge1$ with minimum at $x=1$; for $x\ge2$ and $x\le-1$ the values exceed $3$ or are larger. So the minimum is $f(1)=1$ and (B) is true. Integrating piecewise, $\int_0^1(3-x-x^2)dx=\tfrac{13}{6}$, $\int_1^2(x^2-x+1)dx=\tfrac{11}{6}$, $\int_2^3(x^2+x-3)dx=\tfrac{35}{6}$, total $\tfrac{59}{6}\ne10$; (C) is false. Near $0$, $f=3-x-x^2$, so $f'(0)=-1$, not $1$; (D) is false.