Let `f(x)=abs(x-1)+abs(x-2)+abs(x-3)+abs(x-4),` then

Options

A. least value of f(x) is 4
B. least value is not attained at unique point
C. the number of integral solution of f(x)=4 is 2
D. the value of `(f(pi-1)+f(e))/(2f(12/5))` is 1

Answer

To solve the problem, we need to analyze the function \( f(x) = |x - 1| + |x - 2| + |x - 3| + |x - 4| \). This function consists of absolute values, which means it will behave differently depending on the value of \( x \).

### Step-by-Step Solution:

1. **Identify Critical Points**: The critical points where the expression inside the absolute values changes are \( x = 1, 2, 3, 4 \). We will evaluate the function in the intervals defined by these points: \( (-\infty, 1) \), \( [1, 2) \), \( [2, 3) \), \( [3, 4) \), and \( [4, \infty) \).

2. **Evaluate \( f(x) \) in Each Interval**:
- **For \( x < 1 \)**:
\[ f(x) = -(x - 1) - (x - 2) - (x - 3) - (x - 4) = -4x + 10 \]
- **For \( 1 \leq x < 2 \)**:
\[ f(x) = (x - 1) - (x - 2) - (x - 3) - (x - 4) = -2x + 8 \]
- **For \( 2 \leq x < 3 \)**:
\[ f(x) = (x - 1) + (x - 2) - (x - 3) - (x - 4) = 2x - 6 \]
- **For \( 3 \leq x < 4 \)**:
\[ f(x) = (x - 1) + (x - 2) + (x - 3) - (x - 4) = 4x - 10 \]
- **For \( x \geq 4 \)**:
\[ f(x) = (x - 1) + (x - 2) + (x - 3) + (x - 4) = 4x - 10 \]

3. **Calculate Values at Critical Points**:
- \( f(1) = |1 - 1| + |1 - 2| + |1 - 3| + |1 - 4| = 0 + 1 + 2 + 3 = 6 \)
- \( f(2) = |2 - 1| + |2 - 2| + |2 - 3| + |2 - 4| = 1 + 0 + 1 + 2 = 4 \)
- \( f(3) = |3 - 1| + |3 - 2| + |3 - 3| + |3 - 4| = 2 + 1 + 0 + 1 = 4 \)
- \( f(4) = |4 - 1| + |4 - 2| + |4 - 3| + |4 - 4| = 3 + 2 + 1 + 0 = 6 \)

4. **Determine the Minimum Value**: From the calculations:
- The minimum value of \( f(x) \) occurs at \( x = 2 \) and \( x = 3 \), where \( f(x) = 4 \).

5. **Check the Behavior of \( f(x) \)**:
- For \( x < 1 \): \( f(x) \) is decreasing.
- For \( 1 < x < 2 \): \( f(x) \) is decreasing.
- For \( 2 < x < 3 \): \( f(x) \) is increasing.
- For \( 3 < x < 4 \): \( f(x) \) is increasing.
- For \( x > 4 \): \( f(x) \) is increasing.

### Conclusion:
- The least value of \( f(x) \) is \( 4 \), which occurs at the points \( x = 2 \) and \( x = 3 \).