If `f (x) =2+ |x| -|x-1|-|x+1|,` then `f '((1)/(2)) +f' ((3)/(2))+f' ((5)/(2))` is equal to:

Options

A. 1
B. `-1`
C. 2
D. `-2`

Answer

To solve the problem, we need to find the value of \( f'(\frac{1}{2}) + f'(\frac{3}{2}) + f'(\frac{5}{2}) \) for the function \( f(x) = 2 + |x| - |x-1| - |x+1| \).

### Step 1: Identify the critical points of the function
The critical points for the absolute value functions are:
- \( |x| \) is zero at \( x = 0 \)
- \( |x-1| \) is zero at \( x = 1 \)
- \( |x+1| \) is zero at \( x = -1 \)

Thus, the critical points are \( -1, 0, 1 \).

### Step 2: Determine the intervals
The critical points divide the real line into the following intervals:
1. \( (-\infty, -1) \)
2. \( [-1, 0) \)
3. \( [0, 1) \)
4. \( [1, \infty) \)

### Step 3: Define the function in each interval
1. **For \( x < -1 \)**:
\[ f(x) = 2 + (-x) - (-(x-1)) - (-(x+1)) = 2 - x + x - 1 + x + 1 = 2 \]
Thus, \( f(x) = 2 \).

2. **For \( -1 \leq x < 0 \)**:
\[ f(x) = 2 + (-x) - (-(x-1)) - (x+1) = 2 - x + x - 1 - x - 1 = 0 - x = -x \]

3. **For \( 0 \leq x < 1 \)**:
\[ f(x) = 2 + x - (-(x-1)) - (x+1) = 2 + x + x - 1 - x - 1 = x \]

4. **For \( x \geq 1 \)**:
\[ f(x) = 2 + x - (x-1) - (x+1) = 2 + x - x + 1 - x - 1 = 2 - x \]

### Step 4: Find the derivative in each interval
1. **For \( x < -1 \)**:
\[ f'(x) = 0 \]

2. **For \( -1 \leq x < 0 \)**:
\[ f'(x) = -1 \]

3. **For \( 0 \leq x < 1 \)**:
\[ f'(x) = 1 \]

4. **For \( x \geq 1 \)**:
\[ f'(x) = -1 \]

### Step 5: Evaluate the derivatives at the specified points
- \( f'(\frac{1}{2}) \) lies in the interval \( [0, 1) \), so \( f'(\frac{1}{2}) = 1 \).
- \( f'(\frac{3}{2}) \) lies in the interval \( [1, \infty) \), so \( f'(\frac{3}{2}) = -1 \).
- \( f'(\frac{5}{2}) \) also lies in the interval \( [1, \infty) \), so \( f'(\frac{5}{2}) = -1 \).

### Step 6: Calculate the final result
Now, we can sum these derivatives:
\[ f'(\frac{1}{2}) + f'(\frac{3}{2}) + f'(\frac{5}{2}) = 1 + (-1) + (-1) = 1 - 1 - 1 = -1 \]

### Final Answer
Thus, the value of \( f'(\frac{1}{2}) + f'(\frac{3}{2}) + f'(\frac{5}{2}) \) is \(-1\).

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