Level 2 Calculus Walkthrough

2025 NCEA Level 2 Calculus Question 3(d)

2025 Paper — Find where the function is decreasing

← Back to paper

Question

\[ f(x)=\frac{x^4}{4}+\frac{(k-3)x^3}{3}-\frac{3kx^2}{2}+k \] \[ \text{where }k\text{ is a positive constant.} \]

Determine the regions where \(f(x)\) is decreasing.

Differentiate first, factorise the derivative, find the critical points, use the second derivative test to classify them, then use the sign of the factorised derivative to decide where \(f(x)\) is decreasing.

First walkthrough idea

Tip to try first

Differentiate term by term first, then look for a common factor in the derivative.

Step 1

Differentiate the function

Reveal the explanation and working for this step.

Show the first step’s working

Differentiate the function term by term and collect like powers of \(x\).

Worked result

\[ x^{3} + \left(k - 3\right) x^{2} - 3 k x \]

The derivative of \(-\frac{3kx^2}{2}\) is \(-3kx\), so \(f'(x)=x^3+(k-3)x^2-3kx\).

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91262 — Apply calculus methods in solving problems.

Method: Differentiating, factorising, classifying critical points, and finding where the function is decreasing.

This is Question 3(d) from the 2025 NCEA Level 2 Calculus paper for AS91262 — Apply calculus 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 AS91262.

Page updated .

Learning summary

Review the method, not only the answer

This walkthrough helps you practise differentiating, factorising, classifying critical points, and finding where the function is decreasing. 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 3(c)