| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2020 |
| Session | February |
| Marks | 7 |
| Topic | Tangents, normals and gradients |
| Type | Find second derivative |
| Difficulty | Standard +0.3 This is a straightforward application of finding the second derivative and solving an inequality. Part (a) requires routine differentiation twice (exponential and polynomial terms), then solving a quadratic-like inequality. Part (b) tests basic understanding that d²y/dx² < 0 indicates concavity. While it requires multiple steps and careful algebra, it follows standard procedures without requiring problem-solving insight, making it slightly easier than average. |
| Spec | 1.07e Second derivative: as rate of change of gradient1.07f Convexity/concavity: points of inflection |
6 In this question you must show detailed reasoning.
\begin{enumerate}[label=(\alph*)]
\item Given that $y = \mathrm { e } ^ { 2 x } - 8 \mathrm { e } ^ { x } - 16 x ^ { 2 } + 7 x - 3$, determine the range of values of $x$ for which
$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } < 0$$
\item State the geometrical interpretation of your answer to part (a), in terms of the shape of the graph of $y = \mathrm { f } ( x )$ (not the gradient of the graph).
\end{enumerate}
\hfill \mbox{\textit{SPS SPS SM Pure 2020 Q6 [7]}}