Let [f(x)=2+|x|-|x-1|+|x+1|, x R].

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,

Accepted Answer

$$f(x) = 2 + |x| - |x - 1| + |x + 1|,\,x \in R$$

$$\therefore$$

Solutions · 1