| Exam Board | Edexcel |
|---|---|
| Module | C2 (Core Mathematics 2) |
| Year | 2010 |
| Session | January |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Stationary points and optimisation |
| Type | Second derivative test justification |
| Difficulty | Moderate -0.8 This is a straightforward C2 stationary points question requiring routine differentiation of power functions (including fractional powers), solving dy/dx = 0, finding the second derivative, and applying the second derivative test. All steps are standard textbook procedures with no problem-solving insight needed, making it easier than average but not trivial due to the fractional powers. |
| 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 | Marks | Guidance |
| \(y = 12x^{\frac{1}{2}} - x^{\frac{3}{2}} - 10\) | Given expression | |
| \(y' = 6x^{-\frac{1}{2}} - \frac{3}{2}x^{\frac{1}{2}}\) | M1 A1 | M1 for attempt to differentiate a fractional power \(x^n \to x^{n-1}\); A1 can be unsimplified |
| Puts \(\frac{6}{x^{\frac{1}{2}}} - \frac{3}{2}x^{\frac{1}{2}} = 0\) | M1 | For forming suitable equation using \(y'=0\) |
| \(x = \frac{12}{3} = 4\) (if \(x=0\) also given as solution then lose A1) | M1, A1 | M1 for correct processing of fractional powers leading to \(x = \ldots\); A1 for \(x=4\) only |
| \(x=4 \Rightarrow y = 12 \times 2 - 4^{\frac{3}{2}} - 10\), so \(y=6\) | dM1, A1 (7) | Dependent on three previous M marks; must see evidence of substitution with fractional powers |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| \(y'' = -3x^{-\frac{3}{2}} - \frac{3}{4}x^{-\frac{1}{2}}\) | M1A1 (2) | M1 for differentiating \(y'\) again; A1 should be simplified |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Marks | Guidance |
| Since \(x > 0\), it is a maximum | B1 (1) [10] | Clear conclusion needed; must follow correct \(y''\); dependent on previous A mark; do not need to have found \(x\) earlier |
## Question 9:
### Part (a):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y = 12x^{\frac{1}{2}} - x^{\frac{3}{2}} - 10$ | | Given expression |
| $y' = 6x^{-\frac{1}{2}} - \frac{3}{2}x^{\frac{1}{2}}$ | M1 A1 | M1 for attempt to differentiate a fractional power $x^n \to x^{n-1}$; A1 can be unsimplified |
| Puts $\frac{6}{x^{\frac{1}{2}}} - \frac{3}{2}x^{\frac{1}{2}} = 0$ | M1 | For forming suitable equation using $y'=0$ |
| $x = \frac{12}{3} = 4$ (if $x=0$ also given as solution then lose A1) | M1, A1 | M1 for correct processing of fractional powers leading to $x = \ldots$; A1 for $x=4$ only |
| $x=4 \Rightarrow y = 12 \times 2 - 4^{\frac{3}{2}} - 10$, so $y=6$ | dM1, A1 (7) | Dependent on three previous M marks; must see evidence of substitution with fractional powers |
### Part (b):
| Answer/Working | Marks | Guidance |
|---|---|---|
| $y'' = -3x^{-\frac{3}{2}} - \frac{3}{4}x^{-\frac{1}{2}}$ | M1A1 (2) | M1 for differentiating $y'$ again; A1 should be simplified |
### Part (c):
| Answer/Working | Marks | Guidance |
|---|---|---|
| Since $x > 0$, it is a maximum | B1 (1) **[10]** | Clear conclusion needed; must follow correct $y''$; dependent on previous A mark; do not need to have found $x$ earlier |9. The curve $C$ has equation $y = 12 \sqrt { } ( x ) - x ^ { \frac { 3 } { 2 } } - 10 , \quad x > 0$
\begin{enumerate}[label=(\alph*)]
\item Use calculus to find the coordinates of the turning point on $C$.
\item Find $\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } }$.
\item State the nature of the turning point.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C2 2010 Q9 [10]}}