Level 3 Differentiation Walkthrough
2023 NCEA Level 3 Differentiation Question 2(d)
2023 Paper
Question
Find the \(x\)-value(s) of any points of inflection on the graph of the function.
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.
First walkthrough idea
Hint to try first
Differentiate once with the product rule before finding the second derivative.
Step 1
Differentiate once
Product rule on \(3x^2\ln x\) gives a factorable first derivative.
Show the first step’s working
Product rule on \(3x^2\ln x\) gives a factorable first derivative.
Key result
\[ f'(x)=6x\ln x+3x=3x(1+2\ln x) \]Walkthrough overview
What this question practises
This 2023 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: Second derivatives and finding a logarithmic point of inflection.
This is Question 2(d) from the 2023 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 second derivatives and finding a logarithmic point of inflection. 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.