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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 Explain Question number 6 Adapted from N/A

Question

A function f has its first derivative given by f(x)=(x3)3 .

Find the second derivative.

[2]
a.

Find f(3) and f .

[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 » 6.3 » Points of inflexion with zero and non-zero gradients.
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