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TMUA 2018 · Paper 2 · Question 6 of 20

TMUA 2018 Paper 2 Question 6

Proof and counterexample — Counterexample to a claim about increasing functions. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 2Proof and counterexampleCounterexample to a claim about increasing functions5 options
Which one of the following functions provides a counterexample to the statement:

if f'(x) > 0 for all real x, then f(x) > 0 for all real x.

  1. Af(x) = x2 + 1
  2. Bf(x) = x2 − 1
  3. Cf(x) = x3 + x + 1
  4. Df(x) = 1 − x
  5. Ef(x) = 2x
Show the answer and worked solution
answer · C
  1. Af(x) = x2 + 1
  2. Bf(x) = x2 − 1
  3. Cf(x) = x3 + x + 1
  4. Df(x) = 1 − x
  5. Ef(x) = 2x
A counterexample has to satisfy the hypothesis and fail the conclusion, so it must have f'(x) > 0 everywhere while dipping to zero or below somewhere. The two quadratics have f'(x) = 2x, which is negative for x < 0, and 1 − x has f'(x) = −1, so none of them meets the hypothesis at all. For f(x) = x3 + x + 1, f'(x) = 3x2 + 1 > 0 for every x, yet f(−1) = −1. The exponential does satisfy the hypothesis but is always positive, so it confirms rather than refutes the claim.