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TMUA 2020 · Paper 2 · Question 13 of 20

TMUA 2020 Paper 2 Question 13

Differentiation and integration — Reasoning from an integral equation. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 2Differentiation and integrationReasoning from an integral equation4 options
f(x) is a function for which ∫03 (f(x))2 dx  + ∫03 f(x) dx  =  ∫01 f(x) dx Which of the following claims about f(x) is/are necessarily true?

I   f(x) ≤ 0 for some x with 1 ≤ x ≤ 3
II  ∫03 f(x) dx ≤ ∫01 f(x) dx

  1. Aneither of them
  2. BI only
  3. CII only
  4. DI and II
Show the answer and worked solution
answer · D
  1. Aneither of them
  2. BI only
  3. CII only
  4. DI and II
The one fact to lean on is that (f(x))2 ≥ 0, so ∫03 (f(x))2 dx ≥ 0. Rearranging the given equation, ∫01 f(x) dx − ∫03 f(x) dx = ∫03 (f(x))2 dx ≥ 0, which is exactly statement II. That difference is −∫13 f(x) dx, so ∫13 f(x) dx ≤ 0; a function that were strictly positive across the whole of 1 ≤ x ≤ 3 would give a strictly positive integral there, so f must be zero or negative somewhere in that range, which is statement I. Both are necessarily true.