Cauchy's Mean Value Theorem

Statement: Let $f$ and $g$ be two functions that are continuous on the closed interval $[a, b]$ and differentiable on the open interval $(a, b)$, where $a < b$. Additionally, suppose that $g'(x) \neq 0$ for all $x \in (a, b)$. Then, there exists a point $c \in (a, b)$ such that:

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

Proof:

1. Assumptions:

   • $f$ and $g$ are continuous on $[a, b]$.
   • $f$ and $g$ are differentiable on $(a, b)$.
   • $g'(x) \neq 0$ for all $x \in (a, b)$.

2. Define a new function: We define a function $h(x)$ to apply Rolle's Theorem, which is a key part of the proof. Let:

   $h(x) = f(x) - \lambda g(x)$

   where $\lambda$ is a constant to be determined.

3. Determine $\lambda$: We want to choose $\lambda$ so that $h(a) = h(b)$. This condition will allow us to apply Rolle's Theorem.

   $h(a) = f(a) - \lambda g(a), \quad h(b) = f(b) - \lambda g(b)$

   Setting $h(a) = h(b)$, we get:

   $f(a) - \lambda g(a) = f(b) - \lambda g(b)$

   Solving for $\lambda$, we find:

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

4. Apply Rolle's Theorem: Since $h(a) = h(b)$, by Rolle's Theorem, there exists a point $c \in (a, b)$ such that $h'(c) = 0$. Now, we compute $h'(x)$:

   $h'(x) = f'(x) - \lambda g'(x)$

   Setting $h'(c) = 0$, we get:

   $f'(c) - \lambda g'(c) = 0$

   Substituting the value of $\lambda$, we obtain:

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

5. Conclusion: Therefore, there exists $c \in (a, b)$ such that:

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

This completes the proof of Cauchy's Mean Value Theorem.

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Would you like more details or have any questions about this theorem?

Related Questions:

1. How does Cauchy's Mean Value Theorem differ from the standard Mean Value Theorem?
2. Can you provide an example where Cauchy's Mean Value Theorem is applied?
3. What are the geometric interpretations of Cauchy's Mean Value Theorem?
4. How can we generalize Cauchy's Mean Value Theorem to higher dimensions?
5. How does Rolle's Theorem serve as a foundation for Cauchy's Mean Value Theorem?

Tip:

Always ensure the continuity and differentiability conditions are satisfied before applying the Mean Value Theorems.