| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Classify nature of stationary points |
| Difficulty | Moderate -0.3 This is a straightforward integration problem requiring two successive integrations with boundary conditions. Part (i) is trivial substitution into the second derivative test. Part (ii) involves routine integration of a power function twice, using given conditions to find constants. While it's an 8-mark question requiring multiple steps, each step uses standard techniques with no conceptual challenges or novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| Minimum since \(f''(3) (= 4/3) > 0\) www | B1 | [1] |
| Answer | Marks | Guidance |
|---|---|---|
| \(f'(x) = -18x^{-2}(+c)\) | B1 | |
| \(0 = -2 + c\), \(c = 2\) (\(\rightarrow f'(x) = -18x^{-2} + 2)\) | M1, A1 | Sub \(f'(3) = 0\). (dep c present), \(c = 2\) sufficient at this stage |
| \(f(x) = 18x^{-1} + 2x(+k)\) | B1, B1, M1 | Allow \(cx\) at this stage. Sub \(f(3) = 3\) (k present & numeric (or no) c) |
| \(7 = 6 + 6 + k\), \(k = -5\) (\(\rightarrow f(x) = 18x^{-1} + 2x - 5)\) cao | A1 | [7] |
### (i)
Minimum since $f''(3) (= 4/3) > 0$ www | **B1** | [1]
### (ii)
$f'(x) = -18x^{-2}(+c)$ | **B1** |
$0 = -2 + c$, $c = 2$ ($\rightarrow f'(x) = -18x^{-2} + 2)$ | **M1, A1** | Sub $f'(3) = 0$. (dep c present), $c = 2$ sufficient at this stage
$f(x) = 18x^{-1} + 2x(+k)$ | **B1, B1, M1** | Allow $cx$ at this stage. Sub $f(3) = 3$ (k present & numeric (or no) c)
$7 = 6 + 6 + k$, $k = -5$ ($\rightarrow f(x) = 18x^{-1} + 2x - 5)$ cao | **A1** | [7]
---A curve $y = f(x)$ has a stationary point at $(3, 7)$ and is such that $f''(x) = 36x^{-3}$.
\begin{enumerate}[label=(\roman*)]
\item State, with a reason, whether this stationary point is a maximum or a minimum. [1]
\item Find $f'(x)$ and $f(x)$. [7]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q8 [8]}}