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Rolle’s theorem is a special case of _____
- a)Euclid’s theorem
- b)another form of Rolle’s theorem
- c)Lagrange’s mean value theorem
- d)Joule’s theorem
Correct answer is option 'C'. Can you explain this answer?
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Rolle’s theorem is a special case of _____a)Euclid’s theor...
Rolle’s theorem is just a special case of Lagrange’s mean value theorem when f(a) = f(b) and Lagrange’s mean value theorem is also called the mean value theorem.
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Rolle’s theorem is a special case of _____a)Euclid’s theor...
Understanding Rolle's Theorem
Rolle's Theorem is a fundamental result in calculus, which states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and has equal values at the endpoints (f(a) = f(b)), then there exists at least one point c in (a, b) where the derivative f'(c) = 0.
Connection to Lagrange's Mean Value Theorem
Rolle's Theorem is a specific case of Lagrange's Mean Value Theorem (MVT). The MVT states that if a function is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that:
- f'(c) = (f(b) - f(a)) / (b - a)
Key Points of Lagrange's MVT
- It generalizes Rolle's Theorem by allowing for different function values at the endpoints.
- If f(a) = f(b), then (f(b) - f(a)) = 0, leading to f'(c) = 0, which is precisely the statement of Rolle's Theorem.
Why Option C is Correct
- Since Rolle's Theorem can be derived from Lagrange's MVT by applying the condition f(a) = f(b), it is accurate to say that it is a special case of the Mean Value Theorem.
- This establishes the hierarchy and relationship between these fundamental concepts in calculus.
In summary, understanding Rolle's theorem as a special case of Lagrange's Mean Value Theorem helps in grasping the broader implications of the behavior of functions in calculus.
Rolle's Theorem is a fundamental result in calculus, which states that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b), and has equal values at the endpoints (f(a) = f(b)), then there exists at least one point c in (a, b) where the derivative f'(c) = 0.
Connection to Lagrange's Mean Value Theorem
Rolle's Theorem is a specific case of Lagrange's Mean Value Theorem (MVT). The MVT states that if a function is continuous on [a, b] and differentiable on (a, b), then there exists some c in (a, b) such that:
- f'(c) = (f(b) - f(a)) / (b - a)
Key Points of Lagrange's MVT
- It generalizes Rolle's Theorem by allowing for different function values at the endpoints.
- If f(a) = f(b), then (f(b) - f(a)) = 0, leading to f'(c) = 0, which is precisely the statement of Rolle's Theorem.
Why Option C is Correct
- Since Rolle's Theorem can be derived from Lagrange's MVT by applying the condition f(a) = f(b), it is accurate to say that it is a special case of the Mean Value Theorem.
- This establishes the hierarchy and relationship between these fundamental concepts in calculus.
In summary, understanding Rolle's theorem as a special case of Lagrange's Mean Value Theorem helps in grasping the broader implications of the behavior of functions in calculus.
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Rolle’s theorem is a special case of _____a)Euclid’s theoremb)another form of Rolle’s theoremc)Lagrange’s mean value theoremd)Joule’s theoremCorrect answer is option 'C'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Rolle’s theorem is a special case of _____a)Euclid’s theoremb)another form of Rolle’s theoremc)Lagrange’s mean value theoremd)Joule’s theoremCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Rolle’s theorem is a special case of _____a)Euclid’s theoremb)another form of Rolle’s theoremc)Lagrange’s mean value theoremd)Joule’s theoremCorrect answer is option 'C'. Can you explain this answer?.
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