The differential coefficient of the function `|x-1| +|x-3| ` at the point ` x=2 ` is

Options

A. 2
B. 0
C. `-2`
D. `4`

Answer

To find the differential coefficient of the function \( f(x) = |x - 1| + |x - 3| \) at the point \( x = 2 \), we will follow these steps:

### Step 1: Analyze the function
The function consists of two absolute value terms: \( |x - 1| \) and \( |x - 3| \). We need to determine how these terms behave around \( x = 2 \).

### Step 2: Determine the value of each absolute term at \( x = 2 \)
- For \( |x - 1| \):
\[ |2 - 1| = |1| = 1 \]
- For \( |x - 3| \):
\[ |2 - 3| = |-1| = 1 \]

### Step 3: Write the function without absolute values at \( x = 2 \)
Since \( x = 2 \) is greater than 1 and less than 3, we can express the function without absolute values in this interval:
\[ f(x) = (x - 1) + (3 - x) = 2 \]

### Step 4: Differentiate the function
Now, we differentiate \( f(x) \):
\[ f'(x) = \frac{d}{dx}[(x - 1) + (3 - x)] = \frac{d}{dx}[2] = 0 \]

### Step 5: Evaluate the derivative at \( x = 2 \)
Since the derivative \( f'(x) = 0 \) is constant, the differential coefficient at \( x = 2 \) is:
\[ f'(2) = 0 \]

### Final Answer
The differential coefficient of the function \( |x - 1| + |x - 3| \) at the point \( x = 2 \) is **0**.

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