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Part CCSIR NET December 2023moments-determine-function

Moments determine function

Suppose that f:[−1,1]→Rf : [-1, 1] \to \mathbb{R} is continuous. Which of the following imply that f is identically zero on [−1, 1]?

  1. A.∫\int₋11f(x)xn_{1}^{1} f(x)x^{n} dx = 0 for all n ≥ 0.✓ Correct
  2. B.∫\int₋11f(x)p(x)_{1}^{1} f(x)p(x) dx = 0 for all real polynomials p(x).✓ Correct
  3. C.∫\int₋11f(x)xn_{1}^{1} f(x)x^{n} dx = 0 for all n ≥ 0 odd.
  4. D.∫\int₋11f(x)xn_{1}^{1} f(x)x^{n} dx = 0 for all n ≥ 0 even.

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 hypothesis dropped.

See pricing

50 are analysed free — try those first.

The trap it tests

Hypothesis dropped

A theorem applied without checking one of its preconditions.

Drill statements like this

Related counterexample: |f| Riemann integrable ⇒ f Riemann integrable

More on this topic

The chapter behind this: Riemann integrability — the criterion and the zoo — free to read

From Integration › Riemann integration and criteria

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