Rolle's theorem is a special case of the Lagrange Mean Value Theorem (LMVT). To prove Rolle's Theorem using LMVT, we will begin by stating the two theorems and then show how Rolle's theorem follows from LMVT.

1. Lagrange Mean Value Theorem (LMVT)

Statement:

If a function $f$ is:

1. Continuous on the closed interval $[a, b]$,
2. Differentiable on the open interval $(a, b)$,

then there exists at least one point $c \in (a, b)$ such that:

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

This means that the instantaneous rate of change at some point $c$ is equal to the average rate of change over the interval.

2. Rolle's Theorem

Statement:

If a function $f$ is:

1. Continuous on the closed interval $[a, b]$,
2. Differentiable on the open interval $(a, b)$,
3. $f(a) = f(b)$,

then there exists at least one point $c \in (a, b)$ such that:

$f'(c) = 0.$

3. Proof of Rolle's Theorem Using LMVT

Let $f$ be a function that satisfies the conditions of Rolle's theorem, i.e., it is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$.

From the Lagrange Mean Value Theorem, we know there exists a point $c \in (a, b)$ such that:

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

Now, since $f(a) = f(b)$, the equation simplifies:

$f'(c) = \frac{f(b) - f(a)}{b - a} = \frac{0}{b - a} = 0.$

Thus, $f'(c) = 0$, which is exactly the conclusion of Rolle's Theorem.

Conclusion

Rolle's theorem is a direct consequence of the Lagrange Mean Value Theorem, with the additional condition that $f(a) = f(b)$. The zero slope conclusion follows from this special case, where the average rate of change over the interval is zero.

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Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. What is an example of a function where Rolle’s theorem applies?
2. How does the geometric interpretation of Rolle’s theorem look like?
3. What are the conditions that differentiate Rolle’s theorem from LMVT?
4. Can Rolle's theorem be applied to a function that is not differentiable at some point in the interval?
5. What happens if $f(a) \neq f(b)$? How does LMVT apply then?

Tip: To apply these theorems, always check the continuity and differentiability of the function on the given interval first!