Answer is: option2

Solution:

To determine why the Mean Value Theorem (MVT) does not apply to \( f(x) = |x - 3| \) on the interval \([1, 4]\), let's analyze the conditions of the MVT and the given function:

1. Continuity on \([1, 4]\):
   The function \( f(x) = |x - 3| \) is continuous everywhere, including on the interval \([1, 4]\). Therefore, option A is incorrect.
2. Differentiability on \((1, 4)\):
   The function \( f(x) = |x - 3| \) has a sharp corner at \( x = 3 \), where the derivative does not exist. Since the MVT requires the function to be differentiable on the entire open interval \((1, 4)\), this condition fails. Thus, option B is correct.
3. Equal function values at endpoints (Rolle's Theorem condition):
   The MVT does not require \( f(1) = f(4) \). This is a condition for Rolle’s Theorem, a special case of the MVT. Here, \( f(1) = 2 \) and \( f(4) = 1 \), so \( f(1) \ne f(4) \), but this does not disqualify the MVT. Therefore, option C is irrelevant to the MVT not applying.
4. Function values at endpoints:
   The MVT does not require \( f(1) > f(4) \). This is not a condition of the theorem, so option D is incorrect.

Conclusion:
The Mean Value Theorem does not apply because \( f(x) \) is not differentiable at \( x = 3 \), which lies within the interval \((1, 4)\).

Correct Answer: B