Mean Value Theorem
If f(x) is twice differentiable in [0,2],f(0)=0,f(1)>0 and f(2)=0, then which of the following is necessarily true?
Graph with roots at 0 and 2
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- mean value theorem
- Difficulty
- Hard
- Year
- 2025
Solution
Correct Answer:
f′(c)=0 for some c in (0,2)
Since f(x) is twice differentiable on [0,2], it is continuous on [0,2] and differentiable on (0,2). Given f(0)=0 and f(2)=0: By Rolle's Theorem, there exists at least one c in (0,2) such that f′(c)=0. (While other properties like concavity might be derived from additional data, f′(c)=0 is the direct and necessary consequence of f(0)=f(2)=0).
This hard 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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