Level 2 Calculus Walkthrough

2025 NCEA Level 2 Calculus Question 2(d)

2025 Paper — Find constants from calculus conditions

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Question

\[ f(x)=ax^3+bx^2+ax+2 \]

The graph intersects the x-axis at \(x=1\) and has a stationary point at \(x=-1\).

(i) Find the x-coordinate of the other stationary point.

(ii) Determine what type of stationary point this is.

Differentiate first, use \(f'(-1)=0\) to link \(a\) and \(b\), then use \(f(1)=0\) to solve for the constants before classifying the other stationary point with the second derivative.

First walkthrough idea

Tip to try first

Differentiate first, then use the fact that a stationary point means the derivative is zero.

Step 1

Differentiate the function

Reveal the explanation and working for this step.

Show the first step’s working

Differentiate the function to obtain \(f'(x)\).

Worked result

\[ 3 a x^{2} + 2 b x + a \]

Differentiate each term to get \(f'(x)=3ax^2+2bx+a\).

Walkthrough overview

What this question practises

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

Method: Using a stationary point and an x-intercept to find constants, then classifying the other stationary point.

This is Question 2(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.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise using a stationary point and an x-intercept to find constants, then classifying the other stationary point. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

After solving the derivative condition, check the point's nature and answer the conclusion the question actually asks for.

Continue practising

← Question 2(c) Question 3(a) →