Level 3 Differentiation Walkthrough
2022 NCEA Level 3 Differentiation Question 1(e)
2022 Paper - Proving no points of inflection
Question
If \(p\) is a positive real constant, prove that \(y=e^{px^2}\) does not have any points of inflection.
You must use calculus and show any derivatives that you need.
A later walkthrough visual becomes available with JavaScript; the complete question and first learning steps remain below.
First walkthrough idea
Tip to try first
A point of inflection would need the concavity to change, so start by looking at the second derivative.
Step 1
Find the first derivative
Reveal the explanation and working for this step.
Show the first step’s working
Use the chain rule on \(y=e^{px^2}\).
Worked result
Differentiate \(e^{px^2}\) using the chain rule, multiplying by the derivative of \(px^2\).
Walkthrough overview
What this question practises
This 2022 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.
Method: Proving there are no points of inflection.
This is Question 1(e) from the 2022 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 proving there are no points of inflection. Use the hints to plan the method, then repeat the question without hints and check each step.
Common mistake to avoid
Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.