| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Year | 2008 |
| Session | January |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find range where function increasing/decreasing |
| Difficulty | Moderate -0.8 This is a straightforward C1 stationary points question requiring standard differentiation, solving a quadratic, and using the second derivative test. All techniques are routine with no problem-solving insight needed, making it easier than average but not trivial due to the multi-part nature and algebraic manipulation required. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| \[\frac{dy}{dx} = 3x^2 + 2x - 1\] | *M1 | Attempt to differentiate (at least one correct term). 3 correct terms |
| At stationary points, \(3x^2 + 2x - 1 = 0\) | ||
| \[(3x - 1)(x + 1) = 0\] | DM1 | Correct method to solve 3-term quadratic |
| \[x = \frac{1}{3}, x = -1\] | A1 | |
| \[y = \frac{76}{27}, y = 4\] | A1 6 | |
| SR one correct (x,y) pair www | B1 |
| Answer | Marks | Guidance |
|---|---|---|
| \[\frac{d^2y}{dx^2} = 6x + 2\] | M1 | Looks at sign of \(\frac{d^2y}{dx^2}\) for at least one of their x-values or other correct method |
| \[x = \frac{1}{3}, \frac{d^2y}{dx^2} > 0\] | A1 | \(x = \frac{1}{3}\), minimum point CWO |
| \[x = -1, \frac{d^2y}{dx^2} < 0\] | A1 3 | \(x = -1\), maximum point CWO |
| Answer | Marks | Guidance |
|---|---|---|
| \[-1 < x < \frac{1}{3}\] | M1 | Any inequality (or inequalities) involving both their x values from part (i) |
| A1 2 | Correct inequality (allow \(<\) or \(\leq\)) |
## 8(i)
$$\frac{dy}{dx} = 3x^2 + 2x - 1$$ | *M1 | Attempt to differentiate (at least one correct term). 3 correct terms |
At stationary points, $3x^2 + 2x - 1 = 0$ | | |
$$(3x - 1)(x + 1) = 0$$ | DM1 | Correct method to solve 3-term quadratic |
$$x = \frac{1}{3}, x = -1$$ | A1 | |
$$y = \frac{76}{27}, y = 4$$ | A1 6 | |
**SR** one correct (x,y) pair **www** | B1 |
## 8(ii)
$$\frac{d^2y}{dx^2} = 6x + 2$$ | M1 | Looks at sign of $\frac{d^2y}{dx^2}$ for at least one of their x-values or other correct method |
$$x = \frac{1}{3}, \frac{d^2y}{dx^2} > 0$$ | A1 | $x = \frac{1}{3}$, minimum point CWO |
$$x = -1, \frac{d^2y}{dx^2} < 0$$ | A1 3 | $x = -1$, maximum point CWO |
## 8(iii)
$$-1 < x < \frac{1}{3}$$ | M1 | Any inequality (or inequalities) involving both their x values from part (i) |
| A1 2 | Correct inequality (allow $<$ or $\leq$) |
---8 (i) Find the coordinates of the stationary points on the curve $y = x ^ { 3 } + x ^ { 2 } - x + 3$.\\
(ii) Determine whether each stationary point is a maximum point or a minimum point.\\
(iii) For what values of $x$ does $x ^ { 3 } + x ^ { 2 } - x + 3$ decrease as $x$ increases?
\hfill \mbox{\textit{OCR C1 2008 Q8 [11]}}