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TMUA 2023 · Paper 1 · Question 14 of 20

TMUA 2023 Paper 1 Question 14

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

TMUA 2023 · Paper 1Differentiation and integrationCubics · condition for three distinct roots8 optionshard
The function f(x) = 23x3 + 2mx2 + n,    m > 0 has three distinct real roots.

What is the complete range of possible values of n, in terms of m?

  1. A−83m3 < n < 0
  2. B−43m3 < n < 0
  3. C0 < n < 32m2
  4. D0 < n < 403m3
  5. En < −83m3
  6. Fn < 32m2
  7. Gn > −43m3
  8. Hn > 403m3
Show the answer and worked solution
answer · A
  1. A−83m3 < n < 0
  2. B−43m3 < n < 0
  3. C0 < n < 32m2
  4. D0 < n < 403m3
  5. En < −83m3
  6. Fn < 32m2
  7. Gn > −43m3
  8. Hn > 403m3
f'(x) = 2x2 + 4mx = 2x(x + 2m), so the stationary points are at x = 0 and x = −2m, and since m > 0 the second lies to the left. The leading coefficient is positive, so x = −2m is the local maximum and x = 0 the local minimum. Here f(0) = n and f(−2m) = −163m3 + 8m3 + n = n + 83m3. Three distinct real roots need the local maximum above the axis and the local minimum below it: n + 83 m3 > 0 and n < 0, that is −83 m3 < n < 0.