JEE Math Practice Question

Let $$f(x)=2+|x|-|x-1|+|x+1|, x \in \mathbf{R}$$. Consider $$(\mathrm{S} 1): f^{\prime}\left(-\frac{3}{2}\right)+f^{\prime}\left(-\frac{1}{2}\right)+f^{\prime}\left(\frac{1}{2}\right)+f^{\prime}\left(\frac{3}{2}\right)=2$$ $$(\mathrm{S} 2): \int\limits_{-2}^{2} f(x) \mathrm{d} x=12$$ Then,

1. A.both (S1) and (S2) are correct
2. B.both (S1) and (S2) are wrong
3. C.only (S1) is correct
4. D.only (S2) is correct