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TMUA 2021 · Paper 2 · Question 6 of 20

TMUA 2021 Paper 2 Question 6

Logic and arguments — Necessary and sufficient · roots and stationary points. Try it first; the answer and a full worked solution are below.

TMUA 2021 · Paper 2Logic and argumentsNecessary and sufficient · roots and stationary points4 options
Consider the following two statements about the polynomial f(x):

P:   f(x) = 0 for exactly three real values of x
Q:   f'(x) = 0 for exactly two real values of x

Which one of the following is correct?

  1. AP is necessary but not sufficient for Q.
  2. BP is sufficient but not necessary for Q.
  3. CP is necessary and sufficient for Q.
  4. DP is not necessary and not sufficient for Q.
Show the answer and worked solution
answer · D
  1. AP is necessary but not sufficient for Q.
  2. BP is sufficient but not necessary for Q.
  3. CP is necessary and sufficient for Q.
  4. DP is not necessary and not sufficient for Q.
The cubic case makes both directions look plausible, so the work is in finding polynomials that break each one. For P not sufficient, take f(x) = x4 − x2 = x2(x−1)(x+1): it vanishes at exactly three values −1,  0,  1, but f'(x) = 4x3 − 2x vanishes at three values, not two. For P not necessary, take f(x) = x3 − 3x + 10: f'(x) = 3x2 − 3 vanishes at exactly two values, yet f has a positive minimum (f(1) = 8) so only one real root. Neither implication holds.