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TMUA 2023 · Paper 2 · Question 20 of 20

TMUA 2023 Paper 2 Question 20

Differentiation and integration — Integrals · reasoning about what must be true. Try it first; the answer and a full worked solution are below.

TMUA 2023 · Paper 2Differentiation and integrationIntegrals · reasoning about what must be true8 optionshard
Let f be a polynomial with real coefficients.

The integral Ip, q where p < q is defined by Ip, q = ∫pq (f(x))2 − (f(|x|))2 dx

Which of the following statements must be true?

1   Ip, q = 0 only if 0 < p
2   f'(x) < 0 for all x only if Ip, q < 0 for all p < q < 0
3   Ip, q > 0 only if p < 0

  1. Anone of them
  2. B1 only
  3. C2 only
  4. D3 only
  5. E1 and 2 only
  6. F1 and 3 only
  7. G2 and 3 only
  8. H1, 2 and 3
Show the answer and worked solution
answer · D
  1. Anone of them
  2. B1 only
  3. C2 only
  4. D3 only
  5. E1 and 2 only
  6. F1 and 3 only
  7. G2 and 3 only
  8. H1, 2 and 3
For x ≥ 0 we have |x| = x, so the integrand vanishes there and only the negative part of the range can contribute. Statement 3 follows at once: if p ≥ 0 the whole integral is over non-negative x and equals zero, so Ip, q > 0 forces p < 0. Statement 1 fails for f(x) = x, where the integrand is x2 − x2 = 0 everywhere and Ip, q = 0 for every p. Statement 2 fails for the same f(x) = −x, which is strictly decreasing yet gives Ip, q = 0 rather than a negative value.