| Exam Board | AQA |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Classify nature of stationary points |
| Difficulty | Moderate -0.8 This is a straightforward C1 stationary points question requiring only standard differentiation techniques (power rule), solving a simple equation (x³ = 8), and applying the second derivative test. All parts are routine textbook exercises with no problem-solving insight needed, making it easier than average but not trivial due to the x⁴ term and multiple parts. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx1.07p Points of inflection: using second derivative |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{dy}{dx} = 4x^3 - 32\) | M1, A1, A1 | Reduce one power by 1; One term correct; All correct (no \(+ c\) etc) |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Stationary point \(\Rightarrow \frac{dy}{dx} = 0\) \(\Rightarrow x^3 = 8\) \(\Rightarrow x = 2\) | M1, A1\(\uparrow\), A1 | \(x^n = k\) following from their \(\frac{dy}{dx}\); CSO |
| 3 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{d^2y}{dx^2} = 12x^2\) | B1\(\uparrow\) | FT their \(\frac{dy}{dx}\) |
| 1 |
| Answer | Marks | Guidance |
|---|---|---|
| When \(x = 2\), \(\frac{d^2y}{dx^2}\) considered \(\Rightarrow\) minimum point | M1, E1\(\uparrow\) | Or complete test with \(2 \pm \epsilon\) using \(\frac{dy}{dx}\) |
| 2 |
| Answer | Marks | Guidance |
|---|---|---|
| Putting \(x = 0\) into their \(\frac{dy}{dx}\) (\(= -32\)) \(\frac{dy}{dx} < 0 \Rightarrow\) decreasing | M1, A1\(\uparrow\) | Allow "increasing" if their \(\frac{dy}{dx} > 0\) |
| 2 | ||
| 11 |
## Part (a)
$\frac{dy}{dx} = 4x^3 - 32$ | M1, A1, A1 | Reduce one power by 1; One term correct; All correct (no $+ c$ etc)
| | 3 |
## Part (b)
Stationary point $\Rightarrow \frac{dy}{dx} = 0$ $\Rightarrow x^3 = 8$ $\Rightarrow x = 2$ | M1, A1$\uparrow$, A1 | $x^n = k$ following from their $\frac{dy}{dx}$; CSO
| | 3 |
## Part (c)(i)
$\frac{d^2y}{dx^2} = 12x^2$ | B1$\uparrow$ | FT their $\frac{dy}{dx}$
| | 1 |
## Part (c)(ii)
When $x = 2$, $\frac{d^2y}{dx^2}$ considered $\Rightarrow$ minimum point | M1, E1$\uparrow$ | Or complete test with $2 \pm \epsilon$ using $\frac{dy}{dx}$
| | 2 |
## Part (d)
Putting $x = 0$ into their $\frac{dy}{dx}$ ($= -32$) $\frac{dy}{dx} < 0 \Rightarrow$ decreasing | M1, A1$\uparrow$ | Allow "increasing" if their $\frac{dy}{dx} > 0$
| | 2 |
| | 11 |
---2 The curve with equation $y = x ^ { 4 } - 32 x + 5$ has a single stationary point, $M$.
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Hence find the $x$-coordinate of $M$.
\item \begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
\item Hence, or otherwise, determine whether $M$ is a maximum or a minimum point.
\end{enumerate}\item Determine whether the curve is increasing or decreasing at the point on the curve where $x = 0$.
\end{enumerate}
\hfill \mbox{\textit{AQA C1 2008 Q2 [11]}}