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Questions › DSE › DSE 2018 › Q11
4 marks, −1Mathematics › Limits, continuity and differentiability › Intermediate value theorem, monotonicity and rootsHardabout 3 min

DSE 2018 Question 11: Limits, continuity and differentiability

Let f:[0,1]→Rf : [0,1] \to \mathbb{R}f:[0,1]→R be differentiable and suppose that ∣f′(x)∣<1|f'(x)| < 1∣f′(x)∣<1 for every x∈[0,1]x \in [0,1]x∈[0,1]. Then, there

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Let f: [0,1] → ℝ be differentiable and suppose that |f′(x)| < 1 for every x ∈ [0,1]. Then, there (A) is at least one c ∈ [0,1] such that f(c) = c (B) are two numbers c₁ and c₂ such that f(cᵢ) = cᵢ for i = 1, 2 (C) is exactly one c ∈ [0,1] such that f(c) = c (D) is at most one c ∈ [0,1] such that f(c) = c

Also written as: Let f: [0,1] -> R be differentiable and suppose that |f'(x)| < 1 for every x in [0,1]. Then, there

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 2018 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.