Let's go through the three important theorems in calculus: Rolle's Theorem, the Lagrange Mean Value Theorem (LMVT), and the Cauchy Mean Value Theorem (CMVT). Each theorem provides key insights into the behavior of differentiable functions on an interval.

1. Rolle's Theorem

Statement:
Rolle's Theorem states that if a function $f(x)$ satisfies the following three conditions on a closed interval $[a, b]$:

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 point $c$ in the open interval $(a, b)$ such that:

$f'(c) = 0$

Interpretation:
Rolle's Theorem guarantees that if a function starts and ends at the same value over a closed interval and is smooth and continuous, then there is at least one point where the slope of the tangent (the derivative) is zero within that interval.

2. Lagrange Mean Value Theorem (LMVT)

Statement:
The Lagrange Mean Value Theorem, often referred to simply as the Mean Value Theorem (MVT), states that if a function $f(x)$ satisfies the following two conditions on a closed interval $[a, b]$:

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 point $c$ in the open interval $(a, b)$ such that:

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

Interpretation:
The LMVT states that for a function that is continuous and differentiable on a closed interval, there exists at least one point where the instantaneous rate of change (the derivative) equals the average rate of change over the interval. Geometrically, this means there is at least one point where the tangent to the curve is parallel to the secant line connecting $(a, f(a))$ and $(b, f(b))$.

3. Cauchy Mean Value Theorem (CMVT)

Statement:
The Cauchy Mean Value Theorem generalizes the Lagrange Mean Value Theorem to two functions. It states that if two functions $f(x)$ and $g(x)$ satisfy the following conditions on a closed interval $[a, b]$:

1. Both $f(x)$ and $g(x)$ are continuous on the closed interval $[a, b]$.
2. Both $f(x)$ and $g(x)$ are differentiable on the open interval $(a, b)$.
3. $g'(x) \neq 0$ for all $x \in (a, b)$.

Then there exists at least one point $c$ in the open interval $(a, b)$ such that:

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

Interpretation:
The CMVT is a more general form of the LMVT and applies when considering the ratio of the derivatives of two functions. It states that there is at least one point in the interval where the ratio of the derivatives of the two functions equals the ratio of the total changes of the functions over the interval.

Summary

• Rolle's Theorem: Assumes the function starts and ends at the same value and guarantees a zero derivative somewhere in between.
• Lagrange Mean Value Theorem (LMVT): Generalizes Rolle's Theorem, guaranteeing that the derivative equals the average rate of change over the interval.
• Cauchy Mean Value Theorem (CMVT): Extends the LMVT to two functions, establishing a relationship between their derivatives.

Tip: When applying these theorems, always check the function's continuity and differentiability on the respective intervals as these are crucial for the conditions of these theorems to hold.