The First Mean Value Theorem, often referred to simply as the Mean Value Theorem (MVT), is a fundamental result in calculus that describes a relationship between the derivative of a function and its average rate of change over an interval.

### Mean Value Theorem (MVT)
**Statement:**
Let \( f(x) \) be a function that satisfies the following two conditions on the closed interval \([a, b]\):
1. **Continuity:** \( f(x) \) is continuous on the closed interval \([a, b]\).
2. **Differentiability:** \( f(x) \) is differentiable on the open interval \((a, b)\).

Then, there exists at least one point \( c \) in the open interval \((a, b)\) such that:
\[
f'(c) = \frac{f(b) - f(a)}{b - a}
\]

### Geometric Interpretation:
The theorem states that there is at least one point \( c \) in the interval \((a, b)\) where the tangent to the curve at \( c \) is parallel to the secant line connecting the endpoints \( (a, f(a)) \) and \( (b, f(b)) \).

### Example:
Suppose \( f(x) = x^2 \) on the interval \([1, 4]\). The function is continuous and differentiable everywhere. Applying the MVT:

\[
\frac{f(4) - f(1)}{4 - 1} = \frac{16 - 1}{3} = 5
\]

The derivative of \( f(x) = x^2 \) is \( f'(x) = 2x \). We need to find \( c \) such that \( 2c = 5 \), so \( c = \frac{5}{2} = 2.5 \). This point \( c = 2.5 \) lies within the interval \((1, 4)\), satisfying the MVT.

Would you like more details or have any questions?

### Related Questions:
1. How does the Mean Value Theorem apply to real-world scenarios?
2. Can a function fail the MVT if it’s continuous but not differentiable?
3. What is the difference between Rolle's Theorem and the Mean Value Theorem?
4. How can the MVT be used to prove inequalities?
5. What are some common misunderstandings about the Mean Value Theorem?

**Tip:** When applying the Mean Value Theorem, always check the conditions of continuity and differentiability first to ensure the theorem is applicable.

Explore the First Mean Value Theorem (MVT) in calculus with a detailed explanation and example. Learn how the MVT relates the derivative of a function to its average rate of change over an interval. Suitable for advanced high school students and above.

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