| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2025 |
| Session | October |
| Marks | 6 |
| Topic | Tangents, normals and gradients |
| Type | Determine nature of stationary points |
| Difficulty | Moderate -0.3 This is a straightforward calculus optimization problem requiring differentiation of powers (including fractional powers), solving a quadratic equation, and using the second derivative test. While it involves multiple steps and fractional indices, these are standard A-level techniques with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives |
The curve C is defined for $x > 0$ and has equation
$$y = 3 - \frac{x}{2} - \frac{1}{3\sqrt{x}}$$
\begin{enumerate}[label=\alph*)]
\item Find the exact $x$-coordinate of the stationary point giving your answer in the form $a^b$ where $a$ and $b$ are rational numbers. [4]
\item Find the nature of the stationary point, justifying your answer. [2]
\end{enumerate}
\hfill \mbox{\textit{SPS SPS FM 2025 Q6 [6]}}