| Exam Board | Edexcel |
|---|---|
| Module | C3 (Core Mathematics 3) |
| Marks | 8 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Product & Quotient Rules |
| Type | Determine increasing/decreasing intervals |
| Difficulty | Standard +0.3 This is a straightforward application of the quotient rule followed by solving a quadratic inequality. Part (a) requires routine differentiation and algebraic simplification. Part (b) involves setting f'(x) > 0 and solving, which is standard C3 material with no novel problem-solving required. Slightly above average difficulty due to the algebraic manipulation needed. |
| Spec | 1.07o Increasing/decreasing: functions using sign of dy/dx1.07q Product and quotient rules: differentiation |
| Answer | Marks |
|---|---|
| \(f'(x) = \frac{2x(4x+1)-(x^2+3)x 4}{(4x+1)^2} = \frac{4x^2+2x-12}{(4x+1)^2}\) | M1 A2 |
| Answer | Marks | Guidance |
|---|---|---|
| \(\frac{4x^2+2x-12}{(4x+1)^2} \geq 0\) | M1 | |
| for \(x \neq -\frac{1}{4}\), \((4x+1)^2 > 0\) \(\therefore 4x^2+2x-12 \geq 0\) | M1 A1 | |
| \(2(2x-3)(x+2) \geq 0\) | M1 | |
| \(x \leq -2\) or \(x \geq \frac{3}{2}\) | A1 | (8) |
**(a)**
$f'(x) = \frac{2x(4x+1)-(x^2+3)x 4}{(4x+1)^2} = \frac{4x^2+2x-12}{(4x+1)^2}$ | M1 A2 |
**(b)**
$\frac{4x^2+2x-12}{(4x+1)^2} \geq 0$ | M1 |
for $x \neq -\frac{1}{4}$, $(4x+1)^2 > 0$ $\therefore 4x^2+2x-12 \geq 0$ | M1 A1 |
$2(2x-3)(x+2) \geq 0$ | M1 |
$x \leq -2$ or $x \geq \frac{3}{2}$ | A1 | (8)
---3.
$$f ( x ) = \frac { x ^ { 2 } + 3 } { 4 x + 1 } , \quad x \in \mathbb { R } , \quad x \neq - \frac { 1 } { 4 }$$
\begin{enumerate}[label=(\alph*)]
\item Find and simplify an expression for $\mathrm { f } ^ { \prime } ( x )$.
\item Find the set of values of $x$ for which $\mathrm { f } ( x )$ is increasing.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C3 Q3 [8]}}