If `f(x) = |x-7| + |x -10|+ |x-12|` has a minimum at x=k, then the value of k is _____

Answer

To find the value of \( k \) where the function \( f(x) = |x-7| + |x-10| + |x-12| \) has a minimum, we can follow these steps:

### Step 1: Identify the critical points
The function \( f(x) \) is composed of absolute values, which means it will change its behavior at the points where the expressions inside the absolute values equal zero. The critical points are:
- \( x = 7 \)
- \( x = 10 \)
- \( x = 12 \)

### Step 2: Evaluate the function at the critical points
Next, we will evaluate \( f(x) \) at these critical points.

1. **At \( x = 7 \)**:
\[ f(7) = |7-7| + |7-10| + |7-12| = 0 + 3 + 5 = 8 \]

2. **At \( x = 10 \)**:
\[ f(10) = |10-7| + |10-10| + |10-12| = 3 + 0 + 2 = 5 \]

3. **At \( x = 12 \)**:
\[ f(12) = |12-7| + |12-10| + |12-12| = 5 + 2 + 0 = 7 \]

### Step 3: Compare the values
Now we compare the values of \( f(x) \) at the critical points:
- \( f(7) = 8 \)
- \( f(10) = 5 \)
- \( f(12) = 7 \)

The minimum value occurs at \( x = 10 \) where \( f(10) = 5 \).

### Conclusion
Thus, the value of \( k \) where \( f(x) \) has a minimum is:
\[ \boxed{10} \]