| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2011 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Optimise geometric shape surface area/volume |
| Difficulty | Standard +0.3 This is a standard C2 optimization question requiring product rule differentiation, solving a quadratic equation to find stationary points, and using the second derivative test. While it involves multiple steps, each technique is routine and the problem structure is a textbook exercise with no novel insight required—slightly easier than average. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative |
\begin{enumerate}
\item The volume $V \mathrm {~cm} ^ { 3 }$ of a box, of height $x \mathrm {~cm}$, is given by
\end{enumerate}
$$V = 4 x ( 5 - x ) ^ { 2 } , \quad 0 < x < 5$$
(a) Find $\frac { \mathrm { d } V } { \mathrm {~d} x }$.\\
(b) Hence find the maximum volume of the box.\\
(c) Use calculus to justify that the volume that you found in part (b) is a maximum.
\hfill \mbox{\textit{Edexcel C2 2011 Q10 [10]}}