| Date | May 2015 | Marks available | 4 | Reference code | 15M.1.hl.TZ2.4 |
| Level | HL only | Paper | 1 | Time zone | TZ2 |
| Command term | Determine | Question number | 4 | Adapted from | N/A |
Question
Consider the function defined by \(f(x) = {x^3} - 3{x^2} + 4\).
Determine the values of \(x\) for which \(f(x)\) is a decreasing function.
[4]
a.
There is a point of inflexion, \(P\), on the curve \(y = f(x)\).
Find the coordinates of \(P\).
[3]
b.
Markscheme
attempt to differentiate \(f(x) = {x^3} - 3{x^2} + 4\) M1
\(f'(x) = 3{x^2} - 6x\) A1
\( = 3x(x - 2)\)
(Critical values occur at) \(x = 0,{\text{ }}x = 2\) (A1)
so \(f\) decreasing on \(x \in ]0,{\text{ }}2[\;\;\;({\text{or }}0 < x < 2)\) A1
[4 marks]
a.
\(f''(x) = 6x - 6\) (A1)
setting \(f''(x) = 0\) M1
\( \Rightarrow x = 1\)
coordinate is \((1,{\text{ }}2)\) A1
[3 marks]
Total [7 marks]
b.
Examiners report
[N/A]
a.
[N/A]
b.
