The value of `(d)/(dx)(|x-1|+|x-5|)` at x=3, is

Options

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

Answer

To find the value of \(\frac{d}{dx}(|x-1| + |x-5|)\) at \(x=3\), we will follow these steps:

### Step 1: Analyze the absolute value functions
We need to evaluate the expression \(|x-1| + |x-5|\) at \(x=3\).

- For \(|x-1|\):
- When \(x < 1\), \(|x-1| = 1-x\)
- When \(1 \leq x < 5\), \(|x-1| = x-1\)
- When \(x \geq 5\), \(|x-1| = x-1\)

- For \(|x-5|\):
- When \(x < 5\), \(|x-5| = 5-x\)
- When \(x \geq 5\), \(|x-5| = x-5\)

Since \(x=3\) falls in the range \(1 \leq x < 5\), we can simplify the absolute values:
\[ |x-1| = x-1 \quad \text{and} \quad |x-5| = 5-x \]

### Step 2: Substitute \(x=3\) into the expression
Now we substitute \(x=3\) into the expression:
\[ |3-1| + |3-5| = (3-1) + (5-3) = 2 + 2 = 4 \]

### Step 3: Rewrite the function
Thus, the expression \(|x-1| + |x-5|\) can be rewritten as:
\[ |x-1| + |x-5| = (x-1) + (5-x) = 4 \quad \text{for } 1 \leq x < 5 \]

### Step 4: Differentiate the function
Now we differentiate the constant function \(4\):
\[ \frac{d}{dx}(4) = 0 \]

### Step 5: Conclusion
Therefore, the value of \(\frac{d}{dx}(|x-1| + |x-5|)\) at \(x=3\) is:
\[ \boxed{0} \]