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TMUA 2016 · Paper 2 · Question 18 of 20

TMUA 2016 Paper 2 Question 18

Differentiation and integration — Counterexample to an inequality between integrals. Try it first; the answer and a full worked solution are below.

TMUA 2016 · Paper 2Differentiation and integrationCounterexample to an inequality between integrals6 options
Consider this statement about a function f(x):

(*) If (f(x))2 ≤ 1 for all −1 ≤ x ≤ 1 then ∫−11(f(x))2 dx ≤ ∫−11 f(x) dx

Which one of the following functions provides a counterexample to (*)?

  1. Af(x) = x + 12
  2. Bf(x) = x − 12
  3. Cf(x) = x + x3
  4. Df(x) = x − x3
  5. Ef(x) = x2 + x4
  6. Ff(x) = x2 − x4
Show the answer and worked solution
answer · D
  1. Af(x) = x + 12
  2. Bf(x) = x − 12
  3. Cf(x) = x + x3
  4. Df(x) = x − x3
  5. Ef(x) = x2 + x4
  6. Ff(x) = x2 − x4
A counterexample has to satisfy the hypothesis and break the conclusion, so first throw out anything with (f(x))2 > 1 somewhere on [−1, 1]: at x = 1, option A gives f = 32, option C gives f = 2 and option E gives f = 2, while option B gives f(−1) = −32. That leaves D and F. For f(x) = x − x3 the largest value of |f| on [−1, 1] is 23√3 ≈ 0.385, so the hypothesis holds; but f is odd, so ∫−11 f dx = 0 while ∫−11 f2 dx > 0. The conclusion fails, so D is the counterexample. For F, 0 ≤ f ≤ 14, so f2 ≤ f and the conclusion holds.