To determine if the Mean Value Theorem (MVT) applies to a function $f(x)$ over a given interval $[a, b]$, we need to verify that:

1. The function $f(x)$ is continuous on the closed interval $[a, b]$.
2. The function $f(x)$ is differentiable on the open interval $(a, b)$.

If both conditions are satisfied, then MVT guarantees that there exists at least one point $c \in (a, b)$ such that $f'(c) = \frac{f(b) - f(a)}{b - a}.$

Please provide the function $f(x)$ and the interval $[a, b]$ you are considering, and I can help you determine if the Mean Value Theorem applies.

Would you like further clarification or examples? Here are some related questions:

1. What is the Mean Value Theorem's statement in mathematical terms?
2. How does differentiability differ from continuity?
3. Can you have continuity without differentiability?
4. Are there functions for which the Mean Value Theorem does not apply?
5. How does the Mean Value Theorem relate to the Intermediate Value Theorem?

Tip: When checking if MVT applies, always start by verifying continuity and differentiability over the interval.