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Questions › DSE › DSE 2014 › Q3
1 mark, −0.333333333333333Mathematics › Limits, continuity and differentiability › Intermediate value theorem, monotonicity and rootsModerateabout 2 min

DSE 2014 Question 3: Limits, continuity and differentiability

Let fff be a continuous function from [a,b][a,b][a,b] to [a,b][a,b][a,b], differentiable on (a,b)(a,b)(a,b). A point y∈[a,b]y\in[a,b]y∈[a,b] is a fixed point if y=f(y)y=f(y)y=f(y). If f′(x)≠1f'(x)\neq 1f′(x)=1 for any x∈(a,b)x\in(a,b)x∈(a,b), then fff has

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Let f be a continuous function from [a,b] to [a,b], differentiable on (a,b). A point y∈[a,b] is a fixed point if y=f(y). If f′(x)≠ 1 for any x∈(a,b), then f has (A) multiple, and an odd number of, fixed points (B) no fixed points in [a,b] (C) multiple, but an even number of, fixed points (D) exactly one fixed point in [a,b]

Also written as: Let f be a continuous function from [a,b] to [a,b], differentiable on (a,b). A point y in [a,b] is a fixed point if y=f(y). If f'(x)!= 1 for any x in (a,b), then f has

Limits, continuity and differentiability has come up 122 times in 7 exams since 2006.

What this question tests

This question is on Intermediate value theorem, monotonicity and roots, part of Limits, continuity and differentiability in Mathematics.

Using continuity and the sign of the derivative to show roots exist or are unique.

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Practise Limits, continuity and differentiabilityMore on Intermediate value theorem, monotonicity and roots

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Question from the DSE 2014 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.