The minimum value `|x-6| + |x+3| + |x-8|+|x+4|+|x+3|` is

Options

A. 11
B. 21
C. 31
D. 42

Answer

To find the minimum value of the function \( f(x) = |x - 6| + |x + 3| + |x - 8| + |x + 4| + |x + 3| \), we will follow these steps:

### Step 1: Simplify the Function
We can rewrite the function as:
\[ f(x) = |x - 6| + 2|x + 3| + |x - 8| + |x + 4| \]

### Step 2: Identify Critical Points
The critical points occur where the expressions inside the absolute values change sign. These points are:
- \( x = 6 \)
- \( x = -3 \)
- \( x = 8 \)
- \( x = -4 \)

### Step 3: Evaluate the Function at Critical Points
We will evaluate \( f(x) \) at each of these critical points.

1. **At \( x = 6 \)**:
\[ f(6) = |6 - 6| + 2|6 + 3| + |6 - 8| + |6 + 4| \]
\[ = 0 + 2(9) + 2 + 10 = 0 + 18 + 2 + 10 = 30 \]

2. **At \( x = -3 \)**:
\[ f(-3) = |-3 - 6| + 2|-3 + 3| + |-3 - 8| + |-3 + 4| \]
\[ = 9 + 2(0) + 11 + 1 = 9 + 0 + 11 + 1 = 21 \]

3. **At \( x = 8 \)**:
\[ f(8) = |8 - 6| + 2|8 + 3| + |8 - 8| + |8 + 4| \]
\[ = 2 + 2(11) + 0 + 12 = 2 + 22 + 0 + 12 = 36 \]

4. **At \( x = -4 \)**:
\[ f(-4) = |-4 - 6| + 2|-4 + 3| + |-4 - 8| + |-4 + 4| \]
\[ = 10 + 2(1) + 12 + 0 = 10 + 2 + 12 + 0 = 24 \]

### Step 4: Compare Values
Now we compare the values obtained:
- \( f(6) = 30 \)
- \( f(-3) = 21 \)
- \( f(8) = 36 \)
- \( f(-4) = 24 \)

The minimum value is \( f(-3) = 21 \).

### Conclusion
Thus, the minimum value of the function is:
\[ \boxed{21} \]

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