| Exam Board | Edexcel |
|---|---|
| Module | Paper 2 (Paper 2) |
| Year | 2023 |
| Session | June |
| Marks | 4 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Chain Rule |
| Type | Determine if function is increasing/decreasing |
| Difficulty | Moderate -0.8 This is a straightforward differentiation question requiring two applications of the power rule to find f''(x), solving a linear equation, and interpreting concavity. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial since it requires understanding of second derivatives and concavity. |
| Spec | 1.07f Convexity/concavity: points of inflection1.07i Differentiate x^n: for rational n and sums |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\{f'(x)=\}...x^2+...x+... \Rightarrow \{f''(x)=\}...x+...\) | M1 | For attempting to differentiate twice; valid for \(x^3\to...x^2\to...x\) or \(2x^2\to...x\to k\) or \(-8x\to k\to 0\) |
| \(\{f'(x)=\}3x^2+4x-8 \Rightarrow \{f''(x)=\}6x+4\) | A1cso | Correct second derivative from fully correct work; allow \(6x^1\) for \(6x\) but not \(4x^0\) for 4 unless \(4x^0\) becomes 4 later |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(6x+4=0 \Rightarrow x=-\frac{2}{3}\) | B1ft | ft from solving \(ax+b=0\), \(a,b\neq 0\) where \(ax+b\) is their attempt to differentiate twice; allow equivalent fractions e.g. \(x=-\frac{4}{6}\) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(x\leq -\frac{2}{3}\) or \(x<-\frac{2}{3}\) | B1ft | Deduces \(x\leq -\frac{2}{3}\) or follow through their single value from (i); condone \(<\) for \(\leq\); if 2 inequalities given without indicating which is answer score B0 |
## Question 1:
### Part (a)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\{f'(x)=\}...x^2+...x+... \Rightarrow \{f''(x)=\}...x+...$ | M1 | For attempting to differentiate twice; valid for $x^3\to...x^2\to...x$ or $2x^2\to...x\to k$ or $-8x\to k\to 0$ |
| $\{f'(x)=\}3x^2+4x-8 \Rightarrow \{f''(x)=\}6x+4$ | A1cso | Correct second derivative from fully correct work; allow $6x^1$ for $6x$ but not $4x^0$ for 4 unless $4x^0$ becomes 4 later |
### Part (b)(i)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $6x+4=0 \Rightarrow x=-\frac{2}{3}$ | B1ft | ft from solving $ax+b=0$, $a,b\neq 0$ where $ax+b$ is their attempt to differentiate twice; allow equivalent fractions e.g. $x=-\frac{4}{6}$ |
### Part (b)(ii)
| Answer/Working | Mark | Guidance |
|---|---|---|
| $x\leq -\frac{2}{3}$ or $x<-\frac{2}{3}$ | B1ft | Deduces $x\leq -\frac{2}{3}$ or follow through their single value from (i); condone $<$ for $\leq$; if 2 inequalities given without indicating which is answer score B0 |
---1.
$$f ( x ) = x ^ { 3 } + 2 x ^ { 2 } - 8 x + 5$$
\begin{enumerate}[label=(\alph*)]
\item Find $f ^ { \prime \prime } ( x )$
\item \begin{enumerate}[label=(\roman*)]
\item Solve $\mathrm { f } ^ { \prime \prime } ( x ) = 0$
\item Hence find the range of values of $x$ for which $\mathrm { f } ( x )$ is concave.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel Paper 2 2023 Q1 [4]}}