This problem requires us to work with a piecewise function defined using absolute values. We need to evaluate derivatives and a definite integral of this function. The core skills tested are:

1. Understanding Absolute Value Functions: How to rewrite a function involving absolute values as a piecewise function.
2. Differentiation of Piecewise Functions: Finding the derivative of each piece and evaluating it at specific points.
3. Definite Integration of Piecewise Functions: Splitting the integral into sub-intervals based on the function's definition and integrating each piece.

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1. Defining the Piecewise Function $f(x)$

The given function is $f(x)=2+|x|-|x-1|+|x+1|$. To work with this function, especially for differentiation and integration, we must first express it as a piecewise function by removing the absolute value signs. The critical points where the expressions inside the absolute values change sign are $x=0$ (for $|x|$), $x=1$ (for $|x-1|$), and $x=-1$ (for $|x+1|$). These points divide the real number line into four intervals: $x < -1$, $-1 \le x < 0$, $0 \le x < 1$, and $x \ge 1$.

Let's analyze $f(x)$ in each interval:

• Case 1: $x < -1$ In this interval: $|x| = -x$ (since $x$ is negative) $|x-1| = -(x-1) = -x+1$ (since $x-1$ is negative) $|x+1| = -(x+1) = -x-1$ (since $x+1$ is negative) Substituting these into $f(x)$: $f(x) =