Level 3 Differentiation Walkthrough
2024 NCEA Level 3 Differentiation Question 2(d)
2024 Paper
Question
Consider the function \[ f(x)=\frac{\ln x}{x},\qquad x>0. \]
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.
First walkthrough idea
Hint to try first
At a point of inflection, the second derivative is zero.
Step 1
Find the first derivative
That is the simplified quotient-rule result.
Show the first step’s working
That is the simplified quotient-rule result.
Key result
Walkthrough overview
What this question practises
This 2024 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: Finding a point of inflection on \(\frac{\ln x}{x}\).
This is Question 2(d) from the 2024 NCEA Level 3 Differentiation paper for AS91578 — Apply differentiation methods in solving problems. Use the guided hints to practise the method before revealing the full working.
Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91578.
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Learning summary
Review the method, not only the answer
This walkthrough helps you practise finding a point of inflection on \(\frac{\ln x}{x}\). Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
A zero second derivative alone is not enough; check the required change in concavity or other supporting evidence.