2024 ONE(c)

Merit
Question
For the function below, find the range of values of xx for which the function is decreasing.
y=3(2x−7)2+60ln⁡x+12, x>0y = 3(2x - 7)^2 + 60\ln x + 12,\ x > 0
*You must use calculus and show any derivatives that you need to find when solving this problem.*
Official Answer
Decreasing, therefore

6(2x−7)×2+60x+0<06(2x-7)\times 2+\frac{60}{x}+0<0 (1)

24x−84+60x<024x-84+\frac{60}{x}<0

24x2−84x+60<024x^2-84x+60<0

2x2−7x+5<02x^2-7x+5<0

(2x−5)(x−1)<0(2x-5)(x-1)<0

1<x<2.51<x<2.5
Grading Criteria

Achievement (u)

  • Evidence of reaching stage (1).
  • Can set =0=0 and find the correct critical values for (u).

Merit (r)

  • Finds the correct inequality for required values of xx for decreasing function, with evidence of derivative.

Excellence T1

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Excellence T2

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Video Explanation
2024 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 3:44)
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Exam Paper (2024)Assessment Schedule (2024)
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