| Exam Board | AQA |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Year | 2007 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Find stationary points and nature |
| Difficulty | Standard +0.3 This is a straightforward application of the product rule to find first and second derivatives of a polynomial-exponential product, followed by standard stationary point analysis. While it requires multiple steps and the product rule twice, the algebraic manipulation is routine and the method is a textbook exercise with no novel insight required. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative1.07q Product and quotient rules: differentiation |
7
\begin{enumerate}[label=(\alph*)]
\item A curve has equation $y = \left( x ^ { 2 } - 3 \right) \mathrm { e } ^ { x }$.
\begin{enumerate}[label=(\roman*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$.
\item Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Find the $x$-coordinate of each of the stationary points of the curve.
\item Using your answer to part (a)(ii), determine the nature of each of the stationary points.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{AQA C3 2007 Q7 [10]}}