Continuity
Let f:[0,1]→R be continuous, f(0)=0,f(1)=1. Which of the following is guaranteed by the Intermediate Value Theorem?
f(x) from 0 to 1
Select the correct option:
Solution
f(c)=1−c for some c
We define a new function g(x) to test the condition. Let g(x)=f(x)−(1−x). We know continuous functions on a closed interval satisfy the Intermediate Value Theorem (IVT).
Check values at endpoints: g(0)=f(0)−(1−0)=0−1=−1. g(1)=f(1)−(1−1)=1−0=1.
Since g(0)<0 and g(1)>0, and g is continuous, there must exist at least one c in (0,1) such that g(c)=0. ⟹f(c)−(1−c)=0 ⟹f(c)=1−c.
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About This Question
- Subject
- mathematics
- Chapter
- limit, continuity and differentiability
- Topic
- continuity
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
f(c)=1−c for some c
We define a new function g(x) to test the condition. Let g(x)=f(x)−(1−x). We know continuous functions on a closed interval satisfy the Intermediate Value Theorem (IVT).
Check values at endpoints: g(0)=f(0)−(1−0)=0−1=−1. g(1)=f(1)−(1−1)=1−0=1.
Since g(0)<0 and g(1)>0, and g is continuous, there must exist at least one c in (0,1) such that g(c)=0. ⟹f(c)−(1−c)=0 ⟹f(c)=1−c.
This medium difficulty mathematics question is from the chapter limit, continuity and differentiability, covering the topic of continuity. It appeared in the 2025 exam.
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