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TMUA 2021 · Paper 2 · Question 9 of 20

TMUA 2021 Paper 2 Question 9

Logic and arguments — Sufficient conditions for a real root. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 2Logic and argumentsSufficient conditions for a real root8 options
Consider the following statements about a polynomial f(x):

I    f(x) = px3 + qx2 + rx + s, where p ≠ 0.
II   There is a real number t for which f'(t) = 0.
III There are real numbers u and v for which f(u) f(v) < 0.

Which of these statements is/are sufficient for the equation f(x) = 0 to have a real solution?

  1. AI yes, II yes, III yes
  2. BI yes, II yes, III no
  3. CI yes, II no, III yes
  4. DI yes, II no, III no
  5. EI no, II yes, III yes
  6. FI no, II yes, III no
  7. GI no, II no, III yes
  8. HI no, II no, III no
Show the answer and worked solution
answer · C
  1. AI yes, II yes, III yes
  2. BI yes, II yes, III no
  3. CI yes, II no, III yes
  4. DI yes, II no, III no
  5. EI no, II yes, III yes
  6. FI no, II yes, III no
  7. GI no, II no, III yes
  8. HI no, II no, III no
Take each in turn. I describes a cubic, and a cubic runs from −∞ to +∞ (or the reverse), so it always crosses the axis — sufficient. III says f takes values of opposite signs at u and v; a polynomial is continuous, so by the intermediate value theorem it is zero somewhere between them — sufficient. II is not: f(x) = x2 + 1 has f'(0) = 0 but never reaches zero. So yes, no, yes.