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TMUA 2021 · Paper 2 · Question 12 of 20

TMUA 2021 Paper 2 Question 12

Differentiation and integration — Inequalities under integration and differentiation. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 2Differentiation and integrationInequalities under integration and differentiation8 options
Which of the following statements about polynomials f and g is/are true?

I    If f(x) ≥ g(x) for all x ≥ 0, then ∫0x f(t) dt ≥ ∫0x g(t) dt for all x ≥ 0.
II   If f(x) ≥ g(x) for all x ≥ 0, then f'(x) ≥ g'(x) for all x ≥ 0.
III If f'(x) ≥ g'(x) for all x ≥ 0, then f(x) ≥ g(x) for all x ≥ 0.

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · B
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Work with h = f − g throughout. I says: if h ≥ 0 on [0, x] then ∫0x h ≥ 0 — true, because the integral of a non-negative function over an interval of non-negative length cannot be negative. II asks whether h ≥ 0 forces h' ≥ 0; take h(x) = (x−1)2, which is never negative but has h'(0) = −2. III asks whether h' ≥ 0 forces h ≥ 0; take h(x) = −1, constant, so h' = 0 ≥ 0 while h < 0. Only I.