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Part CCSIR NET June 2025the-derivative-peak-slides-toward-0-without-ever-getting-shorter

The derivative peak slides toward 0 without ever getting shorter

For each n ≥ 1, let fn:R→Rf_{n} : \mathbb{R} \to \mathbb{R} be a function defined by fn(x)=f_{n}(x) = e−n2x2/ne^{-n^{2}x^{2}}/n. Which of the following statements are true?

  1. A.(fn)n≥1(f_{n})_{n}\ge_{1} converges uniformly to 0 on R\mathbb{R}, and (fn′)n≥1(f'_{n})_{n}\ge_{1} converges uniformly to 0 on the interval (−M, M) for some positive real number M.
  2. B.(fn)n≥1(f_{n})_{n}\ge_{1} converges uniformly to 0 on R\mathbb{R}, and (fn′)n≥1(f'_{n})_{n}\ge_{1} converges pointwise to 0 on R\mathbb{R}.✓ Correct
  3. C.(fn)n≥1(f_{n})_{n}\ge_{1} converges uniformly to 0 on R\mathbb{R} and (fn′)n≥1(f'_{n})_{n}\ge_{1} does not converge pointwise to 0 on R\mathbb{R}.
  4. D.(fn)n≥1(f_{n})_{n}\ge_{1} converges pointwise to 0 on R\mathbb{R} but not uniformly on R\mathbb{R}.

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 pointwise vs uniform.

See pricing

50 are analysed free — try those first.

The trap it tests

Pointwise vs uniform

The mode of convergence — or of continuity — was the wrong one.

Drill statements like this

Related counterexample: fₙ → f uniformly ⇒ fₙ′ → f′

More on this topic

The chapter behind this: Uniform convergence — what it buys and how to test it — free to read

From Sequences and Series of Functions › Pointwise vs uniform convergence, M-test, Dini

Last revised . Found a mistake? Tell us — corrections are the fastest thing we act on.

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