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

$$f(x) = \left\{ {\matrix{ {(x - 3)({x^2})} & {3 \le x \le 4} \cr {(x - 3)(2 - {x^2})} & {1 \le x < 3} \cr {(x - 3)({x^2})} & {0 < x < 1} \cr } } \right.$$

$$f'(x) = \left\{ {\matrix{ {3{x^2} - 6x} & {3 < x < 4} \cr { - 3{x^2} + 6x + 2} & {1 < x < 3} \cr {3{x^2} - 6x} & {0 < x < 1} \cr } } \right.$$

$$f'({3^ + }) > 0\,\,\,f'({3^ - }) < 0 \to $$ Minimum

$$f'({1^ + }) > 0\,\,\,f'({1^ - }) < 0 \to $$ Minimum

$$x \in (1,3)\,\,f'(x) = 0$$ at one point $$\to$$ Maximum

$$x \in (3,4)\,\,f'(x) \ne 0$$

$$x \in (0,1)\,\,f'(x) \ne 0$$

So, 3 points.