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TMUA 2018 · Paper 2 · Question 15 of 20

TMUA 2018 Paper 2 Question 15

Graphs and transformations — Cubic graphs · shifting a curve and counting roots. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 2Graphs and transformationsCubic graphs · shifting a curve and counting roots8 optionshard
It is given that f(x) = x3 + 3qx2 + 2, where q is a real constant.

The equation f(x) = 0 has 3 distinct real roots.

Which of the following statements is/are necessarily true?

I    The equation f(x) + 1 = 0 has 3 distinct real roots.
II   The equation f(x+1) = 0 has 3 distinct real roots.
III  The equation f(−x) − 1 = 0 has 3 distinct real roots.

  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Show the answer and worked solution
answer · G
  1. Anone of them
  2. BI only
  3. CII only
  4. DIII only
  5. EI and II only
  6. FI and III only
  7. GII and III only
  8. HI, II and III
Statement II is free: replacing x by x+1 slides the graph sideways, which moves the roots but never changes how many there are. For the other two, locate the turning points. f'(x) = 3x(x + 2q), so the stationary values are f(0) = 2 and f(−2q) = 4q3 + 2. Three distinct roots requires these to straddle zero, so 4q3 + 2 < 0, and then the local maximum is 2 and the local minimum is negative. Statement III asks for f = 1 three times, and 1 always lies strictly between a negative minimum and the maximum 2, so III holds. Statement I asks for f = −1 three times, which needs 4q3 + 2 < −1; taking q = −0.85 satisfies the original condition but gives a minimum of about −0.46, so I can fail.