NCEA Calculus 2024 TWO(d - Solution & Explanation

Question

Consider the function

f(x)=\dfrac{\ln x}{x}

,

x>0

.

Find the coordinates of the point of inflection on the graph of the function.

*You can assume that your point found is actually a point of inflection.*

*You must use calculus and show any derivatives that you need to find when solving this problem.*

Official Answer

f'(x)=\dfrac{1-\ln x}{x^2}

f''(x)=\dfrac{-3+2\ln x}{x^3}

For inflection

f''(x)=0

giving

-3+2\ln x=0

\ln x=\dfrac{3}{2}

x=e^{1.5}=4.4817

y=1.5e^{-1.5}=0.3347

i.e.

(4.4817,0.3347)

Grading Criteria

Achievement (u)

Correct expression for $\dfrac{dy}{dx}$ .

Merit (r)

Correct co-ordinates for point of inflection, with evidence of both $\dfrac{dy}{dx}$ and $\dfrac{d^2y}{dx^2}$ .

Excellence T1

-

Excellence T2

-

Video Explanation

2024 NCEA L3 Calculus Exam Walkthrough by infinityplusone(starts at 27:36)

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2024 TWO(d)

Achievement (u)

Merit (r)

Excellence T1

Excellence T2