| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2020 |
| Session | June |
| Marks | 7 |
| Topic | Factor & Remainder Theorem |
| Type | Multiple unknowns with derivative condition |
| Difficulty | Standard +0.8 This question combines the factor theorem with calculus conditions (inflection point requiring g''(x)=0) to find two unknowns, then requires proving no stationary points exist via discriminant analysis. While systematic, it demands coordination of multiple techniques (factor theorem, second derivative, discriminant) and the inflection point condition is less routine than typical factor theorem questions. |
| Spec | 1.02j Manipulate polynomials: expanding, factorising, division, factor theorem1.07f Convexity/concavity: points of inflection1.07n Stationary points: find maxima, minima using derivatives |
3.
$$\mathrm { g } ( x ) = 4 x ^ { 3 } + a x ^ { 2 } + 4 x + b$$
where $a$ and $b$ are constants.\\
Given that
\begin{itemize}
\item ( $2 x + 1$ ) is a factor of $\mathrm { g } ( x )$
\item the curve with equation $y = \mathrm { g } ( x )$ has a point of inflection at $x = \frac { 1 } { 6 }$
\begin{enumerate}[label=(\alph*)]
\item find the value of $a$ and the value of $b$
\item Show that there are no stationary points on the curve with equation $y = \mathrm { g } ( x )$.
\end{itemize}
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2020 Q3 [7]}}