| Exam Board | OCR |
|---|---|
| Module | C1 (Core Mathematics 1) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Second derivative test justification |
| Difficulty | Moderate -0.3 This is a straightforward stationary points question requiring differentiation of a power function (including fractional power), solving f'(x)=0, and using the second derivative test. All steps are routine C1 techniques with no problem-solving insight needed, making it slightly easier than average, though the fractional power adds minor computational care. |
| Spec | 1.07d Second derivatives: d^2y/dx^2 notation1.07i Differentiate x^n: for rational n and sums1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| \(f'(x) = -1 + 2x^{-\frac{1}{3}}\) | M1 A1 | |
| \(f''(x) = -\frac{2}{3}x^{-\frac{4}{3}}\) | A1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| For TP, \(-1 + 2x^{-\frac{1}{3}} = 0\) | M1 | |
| \(x^{\frac{1}{3}} = 2\) | M1 | |
| \(x = 8 \quad \therefore (8,6)\) | A2 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Notes |
| \(f''(8) = -\frac{1}{24}\), \(f''(x) < 0 \therefore\) maximum | M1 A1 | (9) |
# Question 8:
## Part (i):
| Answer | Mark | Notes |
|--------|------|-------|
| $f'(x) = -1 + 2x^{-\frac{1}{3}}$ | M1 A1 | |
| $f''(x) = -\frac{2}{3}x^{-\frac{4}{3}}$ | A1 | |
## Part (ii):
| Answer | Mark | Notes |
|--------|------|-------|
| For TP, $-1 + 2x^{-\frac{1}{3}} = 0$ | M1 | |
| $x^{\frac{1}{3}} = 2$ | M1 | |
| $x = 8 \quad \therefore (8,6)$ | A2 | |
## Part (iii):
| Answer | Mark | Notes |
|--------|------|-------|
| $f''(8) = -\frac{1}{24}$, $f''(x) < 0 \therefore$ maximum | M1 A1 | **(9)** |
---8.
$$f ( x ) = 2 - x + 3 x ^ { \frac { 2 } { 3 } } , \quad x > 0 .$$
(i) Find $f ^ { \prime } ( x )$ and $f ^ { \prime \prime } ( x )$.\\
(ii) Find the coordinates of the turning point of the curve $y = \mathrm { f } ( x )$.\\
(iii) Determine whether the turning point is a maximum or minimum point.\\
\hfill \mbox{\textit{OCR C1 Q8 [9]}}