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Exam-Style Question on Differentiation

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Question id: 417. This question is similar to one that appeared on an IB AA Standard paper (specimen) for 2021. The use of a calculator is allowed.

Consider the cubic function \(f(x)=\frac{1}{6}x^3-2x^2+6x-2\)

(a) Find \(f'(x)\)

The graph of \(f\) has horizontal tangents at the points where \(x = a\) and \(x = b\) where \( a < b \).

(b) Find the value of \(a\) and the value of \(b\)

(c) Sketch the graph of \(y = f'(x)\).

(d) Hence explain why the graph of \(f\) has a local maximum point at \(x = a\).

(e) Find \(f''(b) \).

(f) Hence, use your answer to part (e) to show that the graph of \(f\) has a local minimum point at \(x = b\).

(g) Find the coordinates of the point where the normal to the graph of \(f\) at \(x = a\) and the tangent to the graph of \(f\) at \(x = b\) intersect.

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The exam-style questions appearing on this site are based on those set in previous examinations (or sample assessment papers for future examinations) by the major examination boards. The wording, diagrams and figures used in these questions have been changed from the originals so that students can have fresh, relevant problem solving practice even if they have previously worked through the related exam paper.

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