TTMUA Lab
TMUA 2018 · Paper 1 · Question 9 of 20

TMUA 2018 Paper 1 Question 9

Differentiation and integration — Cubics · condition for three distinct roots. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 1Differentiation and integrationCubics · condition for three distinct roots6 options
Find the complete set of values of the constant c for which the cubic equation 2x3 − 3x2 − 12x + c = 0 has three distinct real solutions.
  1. A−20 < c < 7
  2. B−7 < c < 20
  3. Cc > 7
  4. Dc > −7
  5. Ec < 20
  6. Fc < −20
Show the answer and worked solution
answer · B
  1. A−20 < c < 7
  2. B−7 < c < 20
  3. Cc > 7
  4. Dc > −7
  5. Ec < 20
  6. Fc < −20
Write f(x) = 2x3 − 3x2 − 12x + c. Then f'(x) = 6x2 − 6x − 12 = 6(x − 2)(x + 1), so the turning points are at x = −1 and x = 2; the positive leading coefficient makes x = −1 the local maximum and x = 2 the local minimum. Three distinct roots need the curve to cross the axis on both sides of each turning point, that is f(−1) > 0 and f(2) < 0. Since f(−1) = 7 + c and f(2) = −20 + c, this gives −7 < c < 20.