# Exam-Style Question on Differentiation

## A mathematics exam-style question with a worked solution that can be revealed gradually

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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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If you need more practice try the self-checking interactive exercises called Differentiation. If you would like to interact with the graph mentioned in this question you will find it waiting for you on the Graph Plotter.

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