| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2011 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Determine constant from stationary point condition |
| Difficulty | Moderate -0.3 This is a straightforward multi-part calculus question testing standard techniques: using the stationary point condition to find a constant, finding the second derivative to classify stationary points, and integrating to recover the original function. All parts follow routine procedures with no problem-solving insight required, making it slightly easier than average but still requiring competent execution of multiple techniques. |
| Spec | 1.07e Second derivative: as rate of change of gradient1.07n Stationary points: find maxima, minima using derivatives1.08a Fundamental theorem of calculus: integration as reverse of differentiation |
| Answer | Marks | Guidance |
|---|---|---|
| \(x = -2, (\) Allow also \(= 3)\) | M1, A1, M1, A1 [4] | AG |
| Answer | Marks | Guidance |
|---|---|---|
| \(f''(-2) < 0\) hence max at \(Q\) | B1, B1 [2] | 3 min, -2 max independent of \(f''(x)\) |
| Answer | Marks | Guidance |
|---|---|---|
| \(c = 17\) | B2,1,0, M1, A1 [4] | Accept anywhere in question; Dependent on c present; Condone \(y =\) or equation \(=\) |
**(i)** $f'(3) = 0 \Rightarrow 18 + 3k - 12 = 0$
$k = -2$
$(x-3)(x+2) = 0$
$x = -2, ($ Allow also $= 3)$ | M1, A1, M1, A1 [4] | AG
**(ii)** $f''(x) = 4x - 2$
$f''(3) > 0$ hence min at $P$
$f''(-2) < 0$ hence max at $Q$ | B1, B1 [2] | 3 min, -2 max independent of $f''(x)$
**(iii)** $f(x) = \frac{2}{3}x^3 - x^2 - 12x (+c)$
Sub $(3, -10) \rightarrow -10 = 18 - 9 - 36 + c$
$c = 17$ | B2,1,0, M1, A1 [4] | Accept anywhere in question; Dependent on c present; Condone $y =$ or equation $=$
---A curve $y = \mathrm{f}(x)$ has a stationary point at $P(3, -10)$. It is given that $\mathrm{f}'(x) = 2x^2 + kx - 12$, where $k$ is a constant.
\begin{enumerate}[label=(\roman*)]
\item Show that $k = -2$ and hence find the $x$-coordinate of the other stationary point, $Q$. [4]
\item Find $\mathrm{f}''(x)$ and determine the nature of each of the stationary points $P$ and $Q$. [2]
\item Find $\mathrm{f}(x)$. [4]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2011 Q8 [10]}}