Find the minimum value of `|x|+|x+1/2|+|x-3|+|x-5/2|dot`

Explanation

To find the minimum value of the function \( f(x) = |x| + |x + \frac{1}{2}| + |x - 3| + |x - \frac{5}{2}| \), we will follow these steps:

### Step 1: Identify the critical points
The critical points occur where the expressions inside the absolute values change sign. This happens at:
- \( x = 0 \)
- \( x = -\frac{1}{2} \)
- \( x = 3 \)
- \( x = \frac{5}{2} \)