Harsh made a piggy bank for himself using clay . If the shape of the bank is based on the function f (x)= | x - 4|+ | x - 5 | where f (x) represents the height of the bank

The value of f'(x) at x = 4 is :

Options

A. `-2`
B. 1
C. `-1`
D. Not defined

Answer

To find the value of \( f'(x) \) at \( x = 4 \) for the function \( f(x) = |x - 4| + |x - 5| \), we will follow these steps:

### Step 1: Understand the function
The function \( f(x) = |x - 4| + |x - 5| \) consists of two absolute value terms. We need to analyze how these terms behave around the point \( x = 4 \).

**Hint:** Recall that the absolute value function behaves differently depending on whether the expression inside is positive or negative.

### Step 2: Determine the intervals
The critical points of the function occur where the expressions inside the absolute values are zero, which are at \( x = 4 \) and \( x = 5 \). We will analyze the function in the intervals:
- \( x < 4 \)
- \( 4 \leq x < 5 \)
- \( x \geq 5 \)

**Hint:** Identify the intervals based on the critical points where the behavior of the function changes.

### Step 3: Define the function in each interval
1. For \( x < 4 \):
\[ f(x) = -(x - 4) - (x - 5) = -x + 4 - x + 5 = -2x + 9 \]

2. For \( 4 \leq x < 5 \):
\[ f(x) = (x - 4) - (x - 5) = x - 4 - x + 5 = 1 \]

3. For \( x \geq 5 \):
\[ f(x) = (x - 4) + (x - 5) = x - 4 + x - 5 = 2x - 9 \]

**Hint:** Write down the piecewise function based on the intervals you identified.

### Step 4: Find the derivative in each interval
1. For \( x < 4 \):
\[ f'(x) = -2 \]

2. For \( 4 \leq x < 5 \):
\[ f'(x) = 0 \]

3. For \( x \geq 5 \):
\[ f'(x) = 2 \]

**Hint:** Differentiate each piece of the piecewise function to find the derivative.

### Step 5: Evaluate \( f'(x) \) at \( x = 4 \)
At \( x = 4 \), we are in the interval \( 4 \leq x < 5 \), where \( f'(x) = 0 \).

**Hint:** Check which interval \( x = 4 \) falls into to determine the correct derivative value.

### Final Answer
The value of \( f'(4) \) is \( 0 \).