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image_path = "/mnt/data/1733531481393wjnhzlvf.jpg" img = Image.open(image_path) img.show()It seems I cannot display the image directly in this environment. I will analyze it to identify the mathematical content, and then I'll proceed to explain or solve it. Let me extract some details from the image now. pytesseract

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extracted_text = pytesseract.image_to_string(img)

extracted_textThe image contains a description related to Rolle’s Theorem. Here's a summarized version of the extracted content:

Rolle’s Theorem states that for a function $f(z)$ defined over a closed interval $[a, b]$, the following conditions must be met:

1. The function $f(z)$ must be continuous on the closed interval $[a, b]$.
2. The function $f(z)$ must be differentiable on the open interval $(a, b)$.
3. The function values at the endpoints must be equal, i.e., $f(a) = f(b)$.

If all of these conditions are satisfied, then there exists a value $c$ in the open interval $(a, b)$ such that: $f'(c) = 0$ This means that at some point within the interval, the derivative (the slope of the function) is zero.

Multiple-Choice Question:

The image also has a multiple-choice question, but it seems incomplete. It appears to be asking about a condition that must be satisfied for Rolle's Theorem. From the visible options, here’s a likely interpretation:

• A: $f(b) = 0$
• B: $f(a) = f(b)$
• C: $f(b) < f(a)$
• D: $f(a) = 0$
• E: $f(a) < f(b)$

The correct answer for Rolle's Theorem is B: $f(a) = f(b)$.

Would you like further explanations on this topic or help with solving related problems?

Related Questions:

1. How does Rolle's Theorem apply to different types of functions (e.g., quadratic, trigonometric)?
2. What is the significance of the condition $f(a) = f(b)$ in Rolle's Theorem?
3. How is the concept of differentiability important in Rolle’s Theorem?
4. What happens if the conditions of Rolle's Theorem are not satisfied?
5. Can Rolle's Theorem be extended to functions defined on unbounded intervals?

Tip:

When solving problems related to Rolle's Theorem, always check for the continuity and differentiability of the function over the given intervals first.