| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2024 |
| Session | October |
| Marks | 6 |
| Topic | Differentiation from First Principles |
| Type | First principles: polynomial with multiple terms |
| Difficulty | Moderate -0.8 This is a straightforward multi-part question testing basic calculus concepts. Part (a) is a standard first-principles differentiation exercise with a simple polynomial. Part (b) requires finding a stationary point and evaluating—routine optimization. Part (c) tests understanding that non-injective functions lack inverses, which is conceptual recall. All parts are textbook-standard with no problem-solving or novel insight required, making this easier than average. |
| Spec | 1.02v Inverse and composite functions: graphs and conditions for existence1.07g Differentiation from first principles: for small positive integer powers of x1.07n Stationary points: find maxima, minima using derivatives |
Given the function $f(x) = x - x^2$, defined for all real values of $x$,
\begin{enumerate}[label=(\alph*)]
\item Show that $f'(x) = 1 - 2x$ by differentiating $f(x)$ from first principles.
[4]
\item Find the maximum value of $f(x)$.
[1]
\item Explain why $f^{-1}(x)$ does not exist.
[1]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2024 Q1 [6]}}