Mean Value Theorem
Easymathematics
Rolle's theorem is applicable to f(x) on [a, b] if
Select the correct option:
Solution
Incorrect! Answer:
f(a) = f(b)
For Rolle's Theorem to be applicable on [a,b], the following conditions must be satisfied:
- f(x) is continuous on the closed interval [a,b].
- f(x) is differentiable on the open interval (a,b).
- The function values at endpoints must be equal: f(a)=f(b).
- Conclusion: If these hold, there exists at least one c∈(a,b) such that f′(c)=0 (the tangent is horizontal at c).
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- mean value theorem
- Difficulty
- Easy
- Year
- 2025
Solution
Correct Answer:
f(a) = f(b)
For Rolle's Theorem to be applicable on [a,b], the following conditions must be satisfied:
- f(x) is continuous on the closed interval [a,b].
- f(x) is differentiable on the open interval (a,b).
- The function values at endpoints must be equal: f(a)=f(b).
- Conclusion: If these hold, there exists at least one c∈(a,b) such that f′(c)=0 (the tangent is horizontal at c).
This easy difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of mean value theorem. It appeared in the 2025 exam.
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