| Exam Board | Edexcel |
|---|---|
| Module | C12 (Core Mathematics 1 & 2) |
| Year | 2016 |
| Session | January |
| Marks | 10 |
| 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 multi-part differentiation question requiring standard techniques: differentiate using power rule, solve dy/dx=0 for stationary point, then use second derivative test. The fractional power (5/4) adds minor complexity but this remains a routine textbook exercise with no problem-solving insight required, making it slightly easier than average. |
| 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 derivatives1.07p Points of inflection: using second derivative |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{dy}{dx} = 12\times\frac{5}{4}x^{\frac{1}{4}} - \frac{10}{18}x\) | M1 A1 | M1: attempt to differentiate, power reduced by one \(x^n \to x^{n-1}\) (not just \(1000\to0\)); A1: two correct terms, no extra terms; may be unsimplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Put \(12\times\frac{5}{4}x^{\frac{1}{4}} - \frac{10}{18}x = 0\) so \(x^n = k\) (\(n\in\mathbb{R}\), \(k\neq0\)) | M1 | Sets derivative \(= 0\), attempts to solve to obtain \(x^n = k\) where \(n\) is real and \(k\) non-zero |
| \(\therefore x = (\ )^{\frac{4}{3}}\) | dM1 | Correct processing to obtain value of \(x\); dependent on first M1; only awarded for equation of form \(ax^{\frac{1}{4}} - bx = 0\) with correct powers |
| \(\therefore x = 81\) (Ignore \(x=0\) if given as second solution) | A1 | cao |
| So \(y = 12(81)^{\frac{5}{4}} - \frac{5}{18}(81)^2 - 1000\) i.e. \(y = 93.5\) | dM1 A1 | dM1: substitutes positive \(x\) into \(y=...\) not into \(\frac{dy}{dx}=...\); A1: cao |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\frac{d^2y}{dx^2} = -\frac{15}{4}x^{-\frac{3}{4}} - \frac{5}{9}\) | B1ft | Correct follow through second derivative |
| Substitutes their non-zero \(x\) (positive or negative) into second derivative | M1 | Note: solving \(\frac{d^2y}{dx^2}=0\) is M0 |
| Obtains negative quantity \(-\frac{15}{36}\) o.e. and considers negative sign deducing maximum | A1 | Completely correct work \((-\frac{5}{12}\) o.e.); must be correct for final mark; e.g. \(\frac{15}{4}\times\frac{1}{27}-\frac{5}{9}\) or \(\frac{15}{108}-\frac{5}{9}\) or \(\frac{5}{36}-\frac{5}{9}\) or \(-0.4...\); correct second derivative followed by \(x=81 \Rightarrow \frac{d^2y}{dx^2} = \frac{15}{4}81^{-\frac{3}{4}}-\frac{5}{9} = -\frac{5}{12}\) scores B1M1A0 |
## Question 10:
### Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{dy}{dx} = 12\times\frac{5}{4}x^{\frac{1}{4}} - \frac{10}{18}x$ | M1 A1 | M1: attempt to differentiate, power reduced by one $x^n \to x^{n-1}$ (not just $1000\to0$); A1: two correct terms, no extra terms; may be unsimplified |
### Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Put $12\times\frac{5}{4}x^{\frac{1}{4}} - \frac{10}{18}x = 0$ so $x^n = k$ ($n\in\mathbb{R}$, $k\neq0$) | M1 | Sets derivative $= 0$, attempts to solve to obtain $x^n = k$ where $n$ is real and $k$ non-zero |
| $\therefore x = (\ )^{\frac{4}{3}}$ | dM1 | Correct processing to obtain value of $x$; dependent on first M1; only awarded for equation of form $ax^{\frac{1}{4}} - bx = 0$ with correct powers |
| $\therefore x = 81$ (Ignore $x=0$ if given as second solution) | A1 | cao |
| So $y = 12(81)^{\frac{5}{4}} - \frac{5}{18}(81)^2 - 1000$ i.e. $y = 93.5$ | dM1 A1 | dM1: substitutes positive $x$ into $y=...$ not into $\frac{dy}{dx}=...$; A1: cao |
### Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\frac{d^2y}{dx^2} = -\frac{15}{4}x^{-\frac{3}{4}} - \frac{5}{9}$ | B1ft | Correct follow through second derivative |
| Substitutes their non-zero $x$ (positive or negative) into second derivative | M1 | Note: solving $\frac{d^2y}{dx^2}=0$ is M0 |
| Obtains negative quantity $-\frac{15}{36}$ o.e. and considers negative sign deducing maximum | A1 | Completely correct work $(-\frac{5}{12}$ o.e.); must be correct for final mark; e.g. $\frac{15}{4}\times\frac{1}{27}-\frac{5}{9}$ or $\frac{15}{108}-\frac{5}{9}$ or $\frac{5}{36}-\frac{5}{9}$ or $-0.4...$; correct second derivative followed by $x=81 \Rightarrow \frac{d^2y}{dx^2} = \frac{15}{4}81^{-\frac{3}{4}}-\frac{5}{9} = -\frac{5}{12}$ scores B1M1A0 |
---10. The curve $C$ has equation
$$y = 12 x ^ { \frac { 5 } { 4 } } - \frac { 5 } { 18 } x ^ { 2 } - 1000 , \quad x > 0$$
\begin{enumerate}[label=(\alph*)]
\item Find $\frac { \mathrm { d } y } { \mathrm {~d} x }$
\item Hence find the coordinates of the stationary point on $C$.
\item Use $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$ to determine the nature of this stationary point.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C12 2016 Q10 [10]}}