| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2014 |
| Session | November |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Find range where function increasing/decreasing |
| Difficulty | Moderate -0.3 This is a straightforward application of differentiation to stationary points. Part (i) requires finding dy/dx = 3x² + 2ax + b and recognizing that no stationary points means the discriminant is negative, leading directly to the required inequality. Part (ii) involves finding where dy/dx < 0 by solving a simple quadratic inequality. Both parts are routine calculus exercises with no novel problem-solving required, making this slightly easier than average. |
| Spec | 1.07n Stationary points: find maxima, minima using derivatives1.07o Increasing/decreasing: functions using sign of dy/dx |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\frac{dy}{dx} = 3x^2 + 2ax + b\) | B1 | co |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow a^2 < 3b\) | M1, A1 [3] | Use of discriminant on their quadratic \(\frac{dy}{dx}\) or other valid method; co – answer given |
| Answer | Marks | Guidance |
|---|---|---|
| \(\rightarrow 1 < x < 3\) | M1, A1, A1 [3] | Attempt at differentiation; co; condone \(\lessgtr\) |
$y = x^3 + ax^2 + bx$
**(i)** $\frac{dy}{dx} = 3x^2 + 2ax + b$ | B1 | co
**(ii)** $b^2 - 4ac = 4a^2 - 12b (< 0)$
$\rightarrow a^2 < 3b$ | M1, A1 [3] | Use of discriminant on their quadratic $\frac{dy}{dx}$ or other valid method; co – answer given
**(iii)** $y = x^3 - 6x^2 + 9x$
$\frac{dy}{dx} = 3x^2 - 12x + 9 < 0$
$= 0$ when $x = 1$ and $3$
$\rightarrow 1 < x < 3$ | M1, A1, A1 [3] | Attempt at differentiation; co; condone $\lessgtr$The equation of a curve is $y = x^3 + ax^2 + bx$, where $a$ and $b$ are constants.
\begin{enumerate}[label=(\roman*)]
\item In the case where the curve has no stationary point, show that $a^2 < 3b$. [3]
\item In the case where $a = -6$ and $b = 9$, find the set of values of $x$ for which $y$ is a decreasing function of $x$. [3]
\end{enumerate}
\hfill \mbox{\textit{CAIE P1 2014 Q6 [6]}}