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TMUA 2016 · Paper 1 · Question 13 of 20

TMUA 2016 Paper 1 Question 13

Differentiation and integration — Counting real roots · turning points and sign changes. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 1Differentiation and integrationCounting real roots · turning points and sign changes5 options
How many real roots does the equation 3x5 − 10x3 − 120x + 30 = 0 have?
  1. A1
  2. B2
  3. C3
  4. D4
  5. E5
Show the answer and worked solution
answer · C
  1. A1
  2. B2
  3. C3
  4. D4
  5. E5
A quintic has at most five roots, and the count is fixed by where the turning points sit relative to the axis. Here f'(x) = 15x4 − 30x2 − 120 = 15(x2 − 4)(x2 + 2), so the only stationary points are x = −2 and x = 2. Evaluating, f(−2) = 254 > 0 is a local maximum and f(2) = −194 < 0 is a local minimum. Since f → −∞ as x → −∞ and f → +∞ as x → +∞, the graph crosses the axis once before x = −2, once between the turning points and once after x = 2: 3 real roots.