| Exam Board | CAIE |
|---|---|
| Module | P2 (Pure Mathematics 2) |
| Year | 2011 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Differentiating Transcendental Functions |
| Type | Find stationary points - logarithmic functions |
| Difficulty | Standard +0.3 This is a straightforward application of the product rule to find dy/dx, setting it equal to zero, and using the second derivative test. The product rule with ln x is standard P2/C3 material, and solving 1 + 2ln x = 0 requires only basic logarithm manipulation. Slightly above average difficulty due to the product rule and logarithmic equation, but remains a routine textbook exercise with no novel problem-solving required. |
| Spec | 1.07l Derivative of ln(x): and related functions1.07n Stationary points: find maxima, minima using derivatives1.07p Points of inflection: using second derivative1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Attempt differentiation using product rule | M1 | |
| Obtain \(8x\ln x + 4x\) (a.c.f.) | A1 | |
| Equate first derivative to zero and attempt solution | M1 | |
| Obtain \(0.607\) | A1 | |
| Obtain \(-0.736\) following their \(x\)-coordinate | A1√ | [5] |
| (ii) Use an appropriate method for determining nature of stationary point | M1 | |
| Conclude point is a minimum (with no errors seen, second derivative = 8) | A1 | [2] |
**(i)** Attempt differentiation using product rule | M1 |
Obtain $8x\ln x + 4x$ (a.c.f.) | A1 |
Equate first derivative to zero and attempt solution | M1 |
Obtain $0.607$ | A1 |
Obtain $-0.736$ following their $x$-coordinate | A1√ | [5]
**(ii)** Use an appropriate method for determining nature of stationary point | M1 |
Conclude point is a minimum (with no errors seen, second derivative = 8) | A1 | [2]6 The curve $y = 4 x ^ { 2 } \ln x$ has one stationary point.\\
(i) Find the coordinates of this stationary point, giving your answers correct to 3 decimal places.\\
(ii) Determine whether this point is a maximum or a minimum point.
\hfill \mbox{\textit{CAIE P2 2011 Q6 [7]}}