| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | February |
| Marks | 8 |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Moderate -0.8 This is a straightforward differentiation and stationary points question requiring only standard power rule application (not chain rule despite the topic label), solving a simple equation, and using the second derivative test. The fractional power is routine for A-level, and all steps are mechanical with no problem-solving insight needed. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative |
4. The curve $C$ has equation
$$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\item Hence find the coordinates of the stationary point on $C$.
\item Use $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ to determine the nature of this stationary point.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q4 [8]}}