2014 THREE(c)

Merit
Question
For what values of xx is the function f(x)=5x−xln⁡xf(x) = 5x - x \ln x increasing?

Show any derivatives that you need to find when solving this problem.
Official Answer
f(x)=5x−xln⁡xf(x)=5x-x\ln x

f′(x)=5−ln⁡x−xxf'(x)=5-\ln x-\dfrac{x}{x}
=4−ln⁡x=4-\ln x

Increasing ⇒\Rightarrow f′(x)>0f'(x)>0

4−ln⁡x>04-\ln x>0
ln⁡x<4\ln x<4
x<e4x<e^4
x<54.6x<54.6

But if x≤0x\le0 then ln⁡x\ln x is not defined, so 0<x<54.60<x<54.6
Grading Criteria

Achievement (u)

  • A correct expression for the derivative.

Merit (r)

  • A correct solution.

Excellence T1

-

Excellence T2

-
Exam Paper (2014)Assessment Schedule (2014)
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