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)=(x−3)3 .
Find the second derivative.
Find f′(3) and f″ .
The point P on the graph of f has x-coordinate 3. Explain why P is not a point of inflexion.
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]
f'(3) = 0 , f''(3) = 0 A1 N1
[1 mark]
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]
Examiners report
Many candidates completed parts (a) and (b) successfully.
Many candidates completed parts (a) and (b) successfully.
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.