| Exam Board | Edexcel |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2019 |
| Session | January |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Find stationary points and nature |
| Difficulty | Moderate -0.3 This is a straightforward differentiation question requiring basic power rule application (not chain rule despite the topic label), solving simple equations, and finding second derivatives. The fractional power is routine for P1 level, and both parts involve direct algebraic manipulation with no conceptual challenges or multi-step reasoning. |
| Spec | 1.07i Differentiate x^n: for rational n and sums1.07j Differentiate exponentials: e^(kx) and a^(kx)1.07n Stationary points: find maxima, minima using derivatives |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f'(x) = 5x^{\frac{3}{2}} - 40\) | M1A1 | M1 for reducing power by one on either term: \(x^{\frac{5}{2}} \to x^{\frac{3}{2}}\) or \(-40x \to -40\) |
| Attempts \(5x^{\frac{3}{2}} - 40 = 0 \Rightarrow x^{\frac{3}{2}} = \ldots\) | M1 | Makes \(x^{\frac{3}{2}}\) the subject. f'\((x)\) must be a changed function |
| \(x = 4\) | A1 cao | Do not accept \(\pm 4\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| \(f''(x) = \frac{15}{2}x^{\frac{1}{2}} = 5\) | M1 | Reducing power by one on a term in \(f'(x)\) and setting \(f''(x) = 5\) |
| \(\Rightarrow x^{\frac{1}{2}} = \ldots \Rightarrow x = \ldots\) | M1A1 | Correct method from \(Ax^{\frac{1}{2}} = 5\): makes \(x^{\frac{1}{2}}\) subject and squares, or squares both sides. \(x = \frac{4}{9}\) or exact equivalent |
## Question 6:
### Part (a):
| Answer | Mark | Guidance |
|--------|------|----------|
| $f'(x) = 5x^{\frac{3}{2}} - 40$ | M1A1 | M1 for reducing power by one on either term: $x^{\frac{5}{2}} \to x^{\frac{3}{2}}$ or $-40x \to -40$ |
| Attempts $5x^{\frac{3}{2}} - 40 = 0 \Rightarrow x^{\frac{3}{2}} = \ldots$ | M1 | Makes $x^{\frac{3}{2}}$ the subject. f'$(x)$ must be a changed function |
| $x = 4$ | A1 cao | Do not accept $\pm 4$ |
### Part (b):
| Answer | Mark | Guidance |
|--------|------|----------|
| $f''(x) = \frac{15}{2}x^{\frac{1}{2}} = 5$ | M1 | Reducing power by one on a term in $f'(x)$ and setting $f''(x) = 5$ |
| $\Rightarrow x^{\frac{1}{2}} = \ldots \Rightarrow x = \ldots$ | M1A1 | Correct method from $Ax^{\frac{1}{2}} = 5$: makes $x^{\frac{1}{2}}$ subject and squares, or squares both sides. $x = \frac{4}{9}$ or exact equivalent |
> Note: Solutions based entirely on graphical or numerical methods score no marks.\begin{enumerate}
\item (Solutions based entirely on graphical or numerical methods are not acceptable.)
\end{enumerate}
Given
$$\mathrm { f } ( x ) = 2 x ^ { \frac { 5 } { 2 } } - 40 x + 8 \quad x > 0$$
(a) solve the equation $\mathrm { f } ^ { \prime } ( x ) = 0$\\
(b) solve the equation $\mathrm { f } ^ { \prime \prime } ( x ) = 5$\\
\hfill \mbox{\textit{Edexcel P1 2019 Q6 [7]}}