TTMUA Lab
TMUA 2016 · Paper 2 · Question 11 of 20

TMUA 2016 Paper 2 Question 11

Differentiation and integration — Polynomials with two real roots · what must follow. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Differentiation and integrationPolynomials with two real roots · what must follow8 options
f(x) is a polynomial with real coefficients.

The equation f(x) = 0 has exactly two real roots, x = −p and x = p, where p > 0.

Consider the following three statements:

1   f'(x) = 0 for exactly one value of x between −p and p
2   The area between the curve y = f(x), the x-axis and the lines x = −p and x = p is given by 2∫0p f(x) dx
3   The graph of y = −f(−x) intersects the x-axis at the points x = −p and x = p only

Which of the above statements must be true?

  1. Anone
  2. B1 only
  3. C2 only
  4. D3 only
  5. E1 and 2 only
  6. F1 and 3 only
  7. G2 and 3 only
  8. H1, 2 and 3
Show the answer and worked solution
answer · D
  1. Anone
  2. B1 only
  3. C2 only
  4. D3 only
  5. E1 and 2 only
  6. F1 and 3 only
  7. G2 and 3 only
  8. H1, 2 and 3
Rolle's theorem guarantees at least one stationary point between the roots, not exactly one. Take f(x) = (x2 − p2)(x2 + ε) with ε small and positive: it still has only ± p as real roots, but f'(x) = 2x(2x2 + ε − p2) vanishes three times inside, so statement 1 fails. Statement 2 assumes f is even and of constant sign, neither of which is given. Statement 3 is safe: −f(−x) = 0 exactly when f(−x) = 0, that is when −x = ± p, so the roots are again ± p and nothing else.