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Part BCSIR NET December 2023ratio-f-prime-over-f

Ratio f prime over f

Let f:R→Rf : \mathbb{R} \to \mathbb{R} be defined by f(x)=(1−x)2sin⁡(x2)f(x) = (1 - x)^{2} \sin(x^{2}) for x ∈ (0, 1) and f(x) = 0 otherwise, and let f′ be its derivative. Let S = {c∈R:f′(x)≤cf(x)c \in \mathbb{R} : f'(x) \le c f(x) for all x∈Rx \in \mathbb{R}}. Which one of the following is true?

  1. A.S = ∅✓ Correct
  2. B.S ≠ ∅ and S is a proper subset of (1,∞)(1, \infty)
  3. C.(2,∞)(2, \infty) is a proper subset of S
  4. D.S ∩ (0, 1) ≠ ∅

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward, and why this one tests execution slip.

See pricing

50 are analysed free — try those first.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: Differentiable ⇒ continuously differentiable

More on this topic

The chapter behind this: Differentiation — MVT, Darboux, Taylor and the classic spoilers — free to read

From Continuity and Differentiation › Differentiability, mean value theorems, Taylor, L'Hôpital

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