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Exam-Style Question on DifferentiationA mathematics exam-style question with a worked solution that can be revealed gradually |
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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