| Exam Board | SPS |
|---|---|
| Module | SPS SM Statistics (SPS SM Statistics) |
| Year | 2022 |
| Session | February |
| Marks | 7 |
| Topic | Tangents, normals and gradients |
| Type | Determine nature of stationary points |
| Difficulty | Moderate -0.8 This is a straightforward differentiation question requiring routine application of the power rule twice, followed by standard stationary point analysis. All steps are mechanical with no problem-solving insight needed—significantly easier than average A-level questions. |
| 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 derivatives1.07p Points of inflection: using second derivative |
3. The curve $C$ has equation
$$y = 5 x ^ { 4 } - 24 x ^ { 3 } + 42 x ^ { 2 } - 32 x + 11 \quad x \in \mathbb { R }$$
\begin{enumerate}[label=(\alph*)]
\item Find
\begin{enumerate}[label=(\roman*)]
\item $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\item $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$
\end{enumerate}\item \begin{enumerate}[label=(\roman*)]
\item Verify that $C$ has a stationary point at $x = 1$
\item Show that this stationary point is a point of inflection, giving reasons for your answer.\\[0pt]
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Statistics 2022 Q3 [7]}}