Prove rolle's theorem from langrange mean value theorem

To prove **Rolle's Theorem** using the **Lagrange Mean Value Theorem (LMVT)**, we first need to recall the statements of both theorems.

### **Rolle's Theorem**:
Let \( f(x) \) be a function that satisfies the following conditions:
1. \( f(x) \) is continuous on the closed interval \([a, b]\),
2. \( f(x) \) is differentiable on the open interval \( (a, b) \),
3. \( f(a) = f(b) \).

Then, there exists at least one \( c \in (a, b) \) such that \( f'(c) = 0 \).

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

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

### **Proof of Rolle's Theorem from LMVT**:
We assume the conditions of **Rolle's Theorem**:
- \( f(x) \) is continuous on \([a, b]\),
- \( f(x) \) is differentiable on \( (a, b) \),
- \( f(a) = f(b) \).

Now, we apply the **Lagrange Mean Value Theorem** to \( f(x) \), which is continuous on \([a, b]\) and differentiable on \( (a, b) \). Therefore, by LMVT, there exists some point \( c \in (a, b) \) such that:
\[
f'(c) = \frac{f(b) - f(a)}{b - a}.
\]

Since \( f(a) = f(b) \), the right-hand side of this equation becomes:
\[
f'(c) = \frac{f(b) - f(a)}{b - a} = \frac{0}{b - a} = 0.
\]

Thus, \( f'(c) = 0 \) for some \( c \in (a, b) \), which proves **Rolle's Theorem**.

### Conclusion:
Rolle's Theorem is a special case of the Lagrange Mean Value Theorem when \( f(a) = f(b) \), resulting in \( f'(c) = 0 \) for some \( c \in (a, b) \).

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Would you like more details or have any questions?

Here are 5 questions that build on this explanation:
1. How can we apply Rolle’s Theorem to polynomial functions?
2. What happens if \( f(a) \neq f(b) \)? Does Rolle’s Theorem still apply?
3. Can you give an example of a function that satisfies the conditions of Rolle's Theorem?
4. How does Rolle's Theorem help in finding stationary points of a function?
5. What is the geometrical interpretation of Rolle's Theorem?

**Tip**: Whenever you use Rolle's Theorem, check the boundary conditions carefully to ensure that \( f(a) = f(b) \).

Prove Rolle's theorem from Lagrange mean value theorem

This detailed explanation shows how to prove Rolle's Theorem using the Lagrange Mean Value Theorem. The problem involves key calculus concepts such as differentiability, continuity, and theorems like Rolle's and LMVT. Ideal for undergraduate students studying calculus.

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