Level 3 Differentiation Walkthrough

2024 NCEA Level 3 Differentiation Question 2(e)

2024 Paper

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Question

The graph of the function \[ y=\frac{xe^{3x}}{2x+k}, \] where \(k\) is a non-zero constant, has a single turning point at \(Q\).

Find the \(x\)-coordinate of the point \(Q\).

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

First walkthrough idea

Focus to try first

differentiating to get the turning-point equation, then using the discriminant to force one repeated solution.

Step 1

Differentiate the function

The derivative tidies up to a nice quadratic factor on top.

Show the first step’s working

The derivative tidies up to a nice quadratic factor on top.

Key result

\[ \frac{e^{\left(3 x\right)} \left(6 x^{2} + 3 k x + k\right)}{\left(2 x + k\right)^{2}} \]

Walkthrough overview

What this question practises

This 2024 walkthrough is part of AS91578 — Apply differentiation methods in solving problems.

Method: Using a discriminant condition to force a single turning point.

This is Question 2(e) 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 using a discriminant condition to force a single turning point. 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.

Continue practising

← Question 2(d) Question 3(a) →