| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Topic | Stationary points and optimisation |
| Type | Determine constant from stationary point condition |
| Difficulty | Standard +0.8 This question requires setting up and solving a system of equations from multiple constraints (coefficient condition, passing through origin, stationary point coordinates). While the individual calculus techniques are standard A-level, synthesizing all conditions to find four unknowns and then applying the second derivative test represents solid problem-solving that goes beyond routine exercises. The 7-mark allocation confirms this is more demanding than typical differentiation questions. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative |
A curve has equation $y = g(x)$.
Given that
• $g(x)$ is a cubic expression in which the coefficient of $x^3$ is equal to the coefficient of $x$
• the curve with equation $y = g(x)$ passes through the origin
• the curve with equation $y = g(x)$ has a stationary point at $(2, 9)$
\begin{enumerate}[label=(\alph*)]
\item find $g(x)$, [7]
\item prove that the stationary point at $(2, 9)$ is a maximum. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2021 Q15 [9]}}