| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find stationary point then sketch curve |
| Difficulty | Moderate -0.3 This is a standard C2 stationary points question with routine differentiation, solving a quadratic, and second derivative test. Part (d) adds mild challenge requiring interpretation of the sketch, but overall this follows a well-practiced algorithm with no novel problem-solving required, making it slightly easier than average. |
| Spec | 1.02n Sketch curves: simple equations including polynomials1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| (a) \(f'(x) = 12x - 3x^2\) | M1 A1 | |
| for SP, \(12x - 3x^2 = 0\) | ||
| \(3x(4 - x) = 0\) | M1 | |
| \(x = 0, 4\) | ||
| \(\therefore (0, 2), (4, 34)\) | A2 | |
| (b) \(f''(x) = 12 - 6x\) | M1 | |
| \(f''(0) = 12, f''(x) > 0 \therefore (0, 2)\) minimum | A1 | |
| \(f''(4) = -12, f''(x) < 0 \therefore (4, 34)\) maximum | A1 | |
| (c) Sketch showing: curve with minimum at origin, maximum between x = 0 and x = 4, then decreases | B2 | |
| (d) \(2 < k < 34\) | B1 | (11 marks) |
**(a)** $f'(x) = 12x - 3x^2$ | M1 A1 |
for SP, $12x - 3x^2 = 0$ | |
$3x(4 - x) = 0$ | M1 |
$x = 0, 4$ | |
$\therefore (0, 2), (4, 34)$ | A2 |
**(b)** $f''(x) = 12 - 6x$ | M1 |
$f''(0) = 12, f''(x) > 0 \therefore (0, 2)$ minimum | A1 |
$f''(4) = -12, f''(x) < 0 \therefore (4, 34)$ maximum | A1 |
**(c)** Sketch showing: curve with minimum at origin, maximum between x = 0 and x = 4, then decreases | B2 |
**(d)** $2 < k < 34$ | B1 | (11 marks)7.
$$f ( x ) = 2 + 6 x ^ { 2 } - x ^ { 3 }$$
\begin{enumerate}[label=(\alph*)]
\item Find the coordinates of the stationary points of the curve $y = \mathrm { f } ( x )$.
\item Determine whether each stationary point is a maximum or minimum point.
\item Sketch the curve $y = \mathrm { f } ( x )$.
\item State the set of values of $k$ for which the equation $\mathrm { f } ( x ) = k$ has three solutions.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 Q7 [11]}}