The Mean Value Theorem (MVT) states that if a function $f$ is continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, then there exists at least one point $c$ in the interval $(a, b)$ such that:

$f'(c) = \frac{f(b) - f(a)}{b - a}.$

In other words, the instantaneous rate of change (the derivative) at some point $c$ is equal to the average rate of change of the function over the interval $[a, b]$.

Conditions for the theorem to apply:

1. $f$ is continuous on $[a, b]$.
2. $f$ is differentiable on $(a, b)$.

This theorem guarantees that the slope of the tangent line at some point $c$ is equal to the slope of the secant line connecting the endpoints $(a, f(a))$ and $(b, f(b))$.

Would you like more details or have any questions?

Here are some related questions:

1. What is the geometric interpretation of the Mean Value Theorem?
2. How is the Mean Value Theorem applied in real-world problems?
3. What is the difference between the Mean Value Theorem and Rolle's Theorem?
4. Can the Mean Value Theorem be extended to vector-valued functions?
5. What are some common counterexamples where the Mean Value Theorem does not apply?

Tip: Remember that differentiability implies continuity, but continuity does not necessarily imply differentiability.