The image contains a question about selecting the correct formulation of Cauchy's theorem from multiple options. Cauchy's theorem, in this context, likely refers to Cauchy's Mean Value Theorem (CMVT), which states:

Let $f(x)$ and $g(x)$ be functions defined on a closed interval $[a, b]$. If:

1. $f(x)$ and $g(x)$ are continuous on $[a, b]$,
2. $f(x)$ 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 a point $c \in (a, b)$ such that:

$$$$

Correct Answer

From the provided options, the last one corresponds to the correct statement of Cauchy's Mean Value Theorem. It mentions:

• Two functions $f(x)$ and $g(x)$ are continuous and differentiable.
• $g'(x) \neq 0$ on $(a, b)$, ensuring the theorem's conditions are met.
• It correctly states the conclusion involving $\frac{f'(c)}{g'(c)}$.

Would you like me to elaborate further on Cauchy's Mean Value Theorem or its applications? Here are some follow-up topics we could explore:

1. What is the difference between Cauchy's Mean Value Theorem and Rolle's Theorem?
2. How does the Mean Value Theorem extend to multiple variables?
3. Why is the condition $g'(x) \neq 0$ necessary in Cauchy's theorem?
4. What are some real-world applications of the Mean Value Theorem?
5. How is Cauchy's theorem related to L'Hopital's Rule?

Tip: Memorizing conditions like continuity and differentiability helps ensure accurate application of these theorems!