B

Step 1: Find the function given:
\( f(x) = x^2 - 6x + 8 \)

Step 2: We need to find the derivative:
\( f'(x) = 2x - 6 \)

Step 3: Set the derivative to zero to find critical points:
\( 2x - 6 = 0 \Rightarrow x = 3 \)

Step 4: Check the critical point within the interval \( [2, 4] \). Since \( x = 3 \) lies in this interval, we will evaluate the function:
\( f(2) = 2^2 - 6 \cdot 2 + 8 = 4 - 12 + 8 = 0 \)
\( f(4) = 4^2 - 6 \cdot 4 + 8 = 16 - 24 + 8 = 0 \)
\( f(3) = 3^2 - 6 \cdot 3 + 8 = 9 - 18 + 8 = -1 \)

Step 5: Determine the maximum and minimum values:
- Minimum occurs at \( f(3) = -1 \)
- Value at boundaries \( f(2) = 0 \) and \( f(4) = 0 \)

Step 6: Therefore, the value of \( f(x) \) vanishing at endpoints gives the value of the function when it is zero for the critical points. Hence the answer is \( 0 \). So choose option B.