| Exam Board | SPS |
|---|---|
| Module | SPS SM Mechanics (SPS SM Mechanics) |
| Year | 2022 |
| Session | February |
| Marks | 7 |
| Topic | Tangents, normals and gradients |
| Type | Increasing/decreasing intervals |
| Difficulty | Standard +0.3 This is a straightforward calculus question requiring quotient rule and chain rule differentiation, followed by showing a quadratic numerator is always positive. The algebra is slightly involved but follows standard A-level techniques with no novel insight required. Slightly easier than average due to the guided structure and routine methods. |
| Spec | 1.02k Simplify rational expressions: factorising, cancelling, algebraic division1.07l Derivative of ln(x): and related functions1.07q Product and quotient rules: differentiation |
The function $f$ is defined by
$$f(x) = \frac{(x + 5)(x + 1)}{(x + 4)} - \ln(x + 4) \quad x \in \mathbb{R} \quad x > k$$
\begin{enumerate}[label=(\alph*)]
\item State the smallest possible value of $k$.
[1]
\item Show that
$$f'(x) = \frac{ax^2 + bx + c}{(x + 4)^2}$$
where $a$, $b$ and $c$ are integers to be found.
[4]
\item Hence show that $f$ is an increasing function.
[2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Mechanics 2022 Q9 [7]}}