To find the minimum value of the function \( f(x) = |3 - x| + |2 + x| + |5 - x| \), we will analyze the function by breaking it down into intervals based on the points where the expressions inside the absolute values change sign. The critical points are \( x = -2, 3, \) and \( 5 \).
### Step-by-Step Solution:
1. **Identify Critical Points**:
The critical points where the expressions inside the absolute values change sign are \( x = -2, 3, \) and \( 5 \).
2. **Define Intervals**:
We will consider the following intervals based on the critical points:
- \( (-\infty, -2) \)
- \( [-2, 3) \)
- \( [3, 5) \)
- \( [5, \infty) \)
3. **Evaluate \( f(x) \) in Each Interval**:
- **Interval 1: \( x < -2 \)**
Here, \( 3 - x > 0 \), \( 2 + x < 0 \), and \( 5 - x > 0 \).
Therefore, \( f(x) = (3 - x) + (-(2 + x)) + (5 - x) = 3 - x - 2 - x + 5 - x = 6 - 3x \).
- **Interval 2: \( -2 \leq x < 3 \)**
Here, \( 3 - x > 0 \), \( 2 + x > 0 \), and \( 5 - x > 0 \).
Therefore, \( f(x) = (3 - x) + (2 + x) + (5 - x) = 3 - x + 2 + x + 5 - x = 10 - x \).
- **Interval 3: \( 3 \leq x < 5 \)**
Here, \( 3 - x < 0 \), \( 2 + x > 0 \), and \( 5 - x > 0 \).
Therefore, \( f(x) = (-(3 - x)) + (2 + x) + (5 - x) = -3 + x + 2 + x + 5 - x = 4 + x \).
- **Interval 4: \( x \geq 5 \)**
Here, \( 3 - x < 0 \), \( 2 + x > 0 \), and \( 5 - x < 0 \).
Therefore, \( f(x) = (-(3 - x)) + (2 + x) + (-(5 - x)) = -3 + x + 2 - 5 + x = 2x - 6 \).
4. **Find Minimum Values at Critical Points**:
Now we will evaluate \( f(x) \) at the critical points and at the boundaries of the intervals:
- At \( x = -2 \):
\( f(-2) = |3 - (-2)| + |2 + (-2)| + |5 - (-2)| = |5| + |0| + |7| = 5 + 0 + 7 = 12 \).
- At \( x = 3 \):
\( f(3) = |3 - 3| + |2 + 3| + |5 - 3| = |0| + |5| + |2| = 0 + 5 + 2 = 7 \).
- At \( x = 5 \):
\( f(5) = |3 - 5| + |2 + 5| + |5 - 5| = | -2 | + |7| + |0| = 2 + 7 + 0 = 9 \).
5. **Compare Values**:
The values we computed are:
- \( f(-2) = 12 \)
- \( f(3) = 7 \)
- \( f(5) = 9 \)
The minimum value occurs at \( x = 3 \) where \( f(3) = 7 \).
### Conclusion:
The minimum value of the function \( f(x) \) is \( \boxed{7} \).
