| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2023 |
| Session | September |
| Marks | 8 |
| Topic | Tangents, normals and gradients |
| Type | Find second derivative |
| Difficulty | Challenging +1.2 This is a standard calculus question requiring two differentiations to find the inflection point, then solving a simple equation. While it involves the product rule twice and exponential differentiation, the algebraic manipulation is straightforward (factoring out e^(x/2) and solving a linear equation). The 8 marks reflect the working required rather than conceptual difficulty—this is a methodical textbook exercise testing technique rather than insight. |
| Spec | 1.07h Differentiation from first principles: for sin(x) and cos(x)1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)1.07p Points of inflection: using second derivative |
A curve has equation $y = xe^{\frac{x}{2}}$
Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection. [8 marks]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q6 [8]}}