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TMUA 2018 · Paper 2 · Question 18 of 20

TMUA 2018 Paper 2 Question 18

Differentiation and integration — Convexity · what the second derivative controls. Try it first; the answer and a full worked solution are below.

TMUA 2018 · Paper 2Differentiation and integrationConvexity · what the second derivative controls6 options
f(x) is a polynomial function defined for all real x.

Which of the following is a necessary condition for the inequality f(a) + f(b)2 ≥ f(a+b2) to be true for all real numbers a and b with a < b?

  1. Af(x) ≥ 0 for all real x
  2. Bf'(x) ≥ 0 for all real x
  3. Cf''(x) ≥ 0 for all real x
  4. Df(x) ≤ 0 for all real x
  5. Ef'(x) ≤ 0 for all real x
  6. Ff''(x) ≤ 0 for all real x
Show the answer and worked solution
answer · C
  1. Af(x) ≥ 0 for all real x
  2. Bf'(x) ≥ 0 for all real x
  3. Cf''(x) ≥ 0 for all real x
  4. Df(x) ≤ 0 for all real x
  5. Ef'(x) ≤ 0 for all real x
  6. Ff''(x) ≤ 0 for all real x
The inequality says the midpoint of the chord from (a, f(a)) to (b, f(b)) sits on or above the curve, for every chord — that is exactly what it means for the graph to be convex. For a polynomial, convexity is controlled by the second derivative: f''(x) ≥ 0 everywhere. The conditions on f itself and on f' are neither needed nor enough, as f(x) = x2 − 5 shows: it fails all four of them yet satisfies the inequality.