| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2024 |
| Session | February |
| Marks | 6 |
| Topic | Product & Quotient Rules |
| Type | Show derivative equals given algebraic form |
| Difficulty | Standard +0.3 This is a straightforward quotient rule application followed by algebraic manipulation to match a given form. Part (a) requires routine differentiation and factoring. Part (b) involves setting the derivative to zero and using the discriminant condition, which is a standard A-level technique. The question is slightly above average due to the algebraic manipulation and discriminant analysis, but remains a typical exam question without requiring novel insight. |
| Spec | 1.07q Product and quotient rules: differentiation1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
7.
The function f is defined by
$$f ( x ) = \frac { e ^ { 3 x } } { 4 x ^ { 2 } + k }$$
where $k$ is a positive constant.
\begin{enumerate}[label=(\alph*)]
\item Show that
$$f ^ { \prime } ( x ) = \left( 12 x ^ { 2 } - 8 x + 3 k \right) g ( x )$$
where $\mathrm { g } ( x )$ is a function to be found.
Given that the curve with equation $y = \mathrm { f } ( x )$ has at least one stationary point,
\item find the range of possible values of $k$.
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2024 Q7 [6]}}