2023 TWO(d)

Merit
Question
Find the xx-value(s) of any points of inflection on the graph of the function f(x)=3x2ln⁡(x)f(x) = 3x^2 \ln(x).

You can assume that your point(s) found are actually point(s) of inflection.

You must use calculus and show any derivatives that you need to find when solving this problem.
Official Answer
f′(x)=3x2⋅1x+ln⁡x⋅(6x)f'(x)=3x^2\cdot\frac{1}{x}+\ln x\cdot(6x)
=3x+6xln⁡x=3x+6x\ln x

f′′(x)=3+6x⋅1x+ln⁡x⋅(6)f''(x)=3+6x\cdot\frac{1}{x}+\ln x\cdot(6)
=9+6ln⁡x=9+6\ln x

f′′(x)=0⇒9+6ln⁡x=0f''(x)=0\Rightarrow 9+6\ln x=0
ln⁡x=−1.5\ln x=-1.5
x=e−1.5 or x=0.223x=e^{-1.5}\text{ or }x=0.223
Grading Criteria

Achievement (u)

  • Correct f′(x)f'(x).

Merit (r)

  • Correct f′(x)f'(x).
    AND
    Correct f′′(x)f''(x).
    AND
    Correct xx-value.

Excellence T1

-

Excellence T2

-
Video Explanation
2023 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 37:53)
Subscribe
Exam Paper (2023)Assessment Schedule (2023)
Related Questions
2024 THREE(e)Excellence

The diagram below shows part of the graph of the function $f(x) = e^{-x^2}$, where $x \geq 0$. [DI...

2024 THREE(c)Merit

Find the $x$-value(s) of any stationary point(s) on the graph of the function $f(x)=\dfrac{x^2-5x+4...

2024 THREE(d)Merit

Jamie is doing some baking and pouring the flour to form a conical pile. The height o...

2024 TWO(e)Excellence

The graph of the function $y = \dfrac{xe^{3x}}{2x + k}$, where $k$ is a non-zero constant, has a sin...

2024 TWO(c)Merit

Show that $y = \sin(x^2) - \cos(x)$ is a solution to the equation $\dfrac{d^2y}{dx^2} + 4x^2\,y = 2...