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Date May 2009 Marks available 2 Reference code 09M.1.sl.TZ2.6
Level SL only Paper 1 Time zone TZ2
Command term Find Question number 6 Adapted from N/A

Question

A function f has its first derivative given by \(f'(x) = {(x - 3)^3}\) .

Find the second derivative.

[2]
a.

Find \(f'(3)\) and \(f''(3)\) .

[1]
b.

The point P on the graph of f has x-coordinate \(3\). Explain why P is not a point of inflexion.

[2]
c.

Markscheme

METHOD 1

\(f''(x) = 3{(x - 3)^2}\)     A2     N2

METHOD 2

attempt to expand \({(x - 3)^3}\)     (M1)

e.g. \(f'(x) = {x^3} - 9{x^2} + 27x - 27\)

\(f''(x) = 3{x^2} - 18x + 27\)     A1     N2

[2 marks]

a.

\(f'(3) = 0\) , \(f''(3) = 0\)     A1     N1

[1 mark]

b.

METHOD 1

\({f''}\) does not change sign at P     R1

evidence for this     R1     N0

METHOD 2

\({f'}\) changes sign at P so P is a maximum/minimum (i.e. not inflexion)     R1

evidence for this     R1     N0

METHOD 3

finding \(f(x) = \frac{1}{4}{(x - 3)^4} + c\) and sketching this function     R1

indicating minimum at \(x = 3\)     R1     N0 

[2 marks]

c.

Examiners report

Many candidates completed parts (a) and (b) successfully.

a.

Many candidates completed parts (a) and (b) successfully.

b.

A rare few earned any marks in part (c) - most justifying the point of inflexion with the zero answers in part (b), not thinking that there is more to consider.

c.

Syllabus sections

Topic 6 - Calculus » 6.2 » The second derivative.

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