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TMUA 2020 · Paper 2 · Question 12 of 20

TMUA 2020 Paper 2 Question 12

Differentiation and integration — Trapezium rule · overestimates and concavity. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Differentiation and integrationTrapezium rule · overestimates and concavity6 options
Which one of A–F correctly completes the following statement?

Given that a < b, and f(x) > 0 for all x with a < x < b, the trapezium rule produces an overestimate for ∫ab f(x) dx …

  1. A… if f'(x) > 0 and f''(x) < 0 for all x with a < x < b
  2. B… only if f'(x) > 0 and f''(x) < 0 for all x with a < x < b
  3. C… if and only if f'(x) > 0 and f''(x) < 0 for all x with a < x < b
  4. D… if f'(x) < 0 and f''(x) > 0 for all x with a < x < b
  5. E… only if f'(x) < 0 and f''(x) > 0 for all x with a < x < b
  6. F… if and only if f'(x) < 0 and f''(x) > 0 for all x with a < x < b
Show the answer and worked solution
answer · D
  1. A… if f'(x) > 0 and f''(x) < 0 for all x with a < x < b
  2. B… only if f'(x) > 0 and f''(x) < 0 for all x with a < x < b
  3. C… if and only if f'(x) > 0 and f''(x) < 0 for all x with a < x < b
  4. D… if f'(x) < 0 and f''(x) > 0 for all x with a < x < b
  5. E… only if f'(x) < 0 and f''(x) > 0 for all x with a < x < b
  6. F… if and only if f'(x) < 0 and f''(x) > 0 for all x with a < x < b
What controls the trapezium rule is the second derivative alone: each chord lies above a curve that bends upwards, so f''(x) > 0 throughout gives an overestimate, and f''(x) < 0 gives an underestimate. Whether the function is increasing or decreasing is irrelevant. So any option with f'' < 0 is wrong, ruling out A, B and C. Between the remaining three, f' < 0 and f'' > 0 is enough to guarantee an overestimate, so "if" is correct. It is not "only if": f(x) = ex on [0, 1] is increasing and convex and still overestimated, so the condition is not necessary, and E and F fail.