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TMUA 2017 · Paper 2 · Question 16 of 20

TMUA 2017 Paper 2 Question 16

Differentiation and integration — Counterexample · integer values of a derivative. Try it first; the answer and a full worked solution are below.

TMUA 2017 · Paper 2Differentiation and integrationCounterexample · integer values of a derivative4 options
Consider the following statement:

(×) If f(x) is an integer for every integer x, then f'(x) is an integer for every integer x.

Which one of the following is a counterexample to (×)?

  1. Af(x) = x3 + x + 14
  2. Bf(x) = x4 + x2 + x2
  3. Cf(x) = x4 + x3 + x2 + x2
  4. Df(x) = x4 + 2x3 + x24
Show the answer and worked solution
answer · C
  1. Af(x) = x3 + x + 14
  2. Bf(x) = x4 + x2 + x2
  3. Cf(x) = x4 + x3 + x2 + x2
  4. Df(x) = x4 + 2x3 + x24
A counterexample has to satisfy the hypothesis first, so start by testing small integers. Option A fails at x = 0, giving 14, and option B fails at x = 1, giving 32. In option C the numerator factorises as x(x+1)(x2+1), and x(x+1) is always even, so f(x) is always an integer; but f'(x) = 4x3+3x2+2x+12, and at x = 2 this is 492, which is not. Option D is (x(x+1)2)2, always an integer, and its derivative x(x+1)(2x+1)2 is an integer too, so it satisfies the statement rather than breaking it.