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TMUA 2020 · Paper 1 · Question 6 of 20

TMUA 2020 Paper 1 Question 6

Exponentials and logarithms — Exponentials · maximum of a reciprocal. Try it first; the answer and a full worked solution are below.

TMUA 2020 · Paper 1Exponentials and logarithmsExponentials · maximum of a reciprocal6 options
Find the maximum value of the function f(x) = 152x − 4(5x) + 7
  1. A17
  2. B14
  3. C13
  4. D3
  5. E4
  6. F7
Show the answer and worked solution
answer · C
  1. A17
  2. B14
  3. C13
  4. D3
  5. E4
  6. F7
Substitute t = 5x, which takes every positive value. The denominator becomes t2 − 4t + 7 = (t−2)2 + 3, always positive, with least value 3 at t = 2 — and t = 2 is attainable, at x = log5 2. Since the denominator is positive throughout, f is largest when the denominator is smallest, giving a maximum of 13. Option A comes from setting t = 1 (that is x = 0) rather than minimising.