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TMUA 2017 · Paper 2 · Question 19 of 20

TMUA 2017 Paper 2 Question 19

Differentiation and integration — Cubics · stationary values and the number of roots. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 2Differentiation and integrationCubics · stationary values and the number of roots8 options
Which one of the following is a sufficient condition for the equation x3 − 3x2 + a = 0, where a is a constant, to have exactly one real root?
  1. Aa > 0
  2. Ba ≤ 0
  3. Ca ≥ 4
  4. Da < 4
  5. E|a| > 4
  6. F|a| ≤ 4
  7. Ga = 94
  8. H|a| = 32
Show the answer and worked solution
answer · E
  1. Aa > 0
  2. Ba ≤ 0
  3. Ca ≥ 4
  4. Da < 4
  5. E|a| > 4
  6. F|a| ≤ 4
  7. Ga = 94
  8. H|a| = 32
Let y = x3 − 3x2 + a. Then dydx = 3x2 − 6x = 3x(x−2), giving a local maximum at x = 0 with value a and a local minimum at x = 2 with value a − 4. A cubic has exactly one real root precisely when those two stationary values have the same sign, that is a(a−4) > 0, so a < 0 or a > 4. Now check the options: |a| > 4 means a > 4 or a < −4, and both lie inside that region, so it is sufficient. The near misses fail at their endpoints — a = 0 gives x2(x−3) and a = 4 gives (x+1)(x−2)2, each with two distinct real roots.