Standard +0.3 This is a straightforward stationary point question requiring quotient rule (or rewriting as a product), setting derivative to zero, and solving. While it involves multiple steps and algebraic manipulation with surds, it follows a standard template with no conceptual surprises—slightly easier than average for A-level but not trivial due to the surd manipulation required.
9. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
A curve has equation
$$y = \frac { 4 x ^ { 2 } + 9 } { 2 \sqrt { x } } \quad x > 0$$
Find the \(x\) coordinate of the point on the curve at which \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 0\)
Attempts to divide by \(2\sqrt{x}\). Award for one correct term including \(\dfrac{9}{2x^{\frac{1}{2}}}\). Allow if they combine with common denominator of 2, but indices must be processed.
Attempts to differentiate. Award for one power decreasing by one, indices must be processed. Cannot award if just differentiating top and bottom of fraction separately.
Sets \(\dfrac{dy}{dx} = 0\) and proceeds to \(x^{\pm 2} = \ldots\) or \(x^{\pm 4} = \ldots\) following a derivative of form \(Ax^{\frac{1}{2}} - Bx^{-\frac{3}{2}},\ A,B > 0\)
\(x = \dfrac{\sqrt{3}}{2}\)
A1
Or exact equivalent cso. A correct exact answer can imply final M1A1. Rounded answer with no working (e.g. awrt 0.87) is M0A0. Withhold final mark if \(-\dfrac{\sqrt{3}}{2}\) is not rejected.
## Question 9:
| Answer | Marks | Guidance |
|--------|-------|----------|
| $\dfrac{4x^2 + 9}{2\sqrt{x}} = \dfrac{4x^2}{2\sqrt{x}} + \dfrac{9}{2\sqrt{x}} = 2x^{\frac{3}{2}} + \dfrac{9}{2}x^{-\frac{1}{2}}$ | M1 | Attempts to divide by $2\sqrt{x}$. Award for one correct term including $\dfrac{9}{2x^{\frac{1}{2}}}$. Allow if they combine with common denominator of 2, but indices must be processed. |
| $2x^{\frac{3}{2}} + \dfrac{9}{2}x^{-\frac{1}{2}}$ | A1 | May be left unsimplified but indices must be processed |
| $\dfrac{dy}{dx} = 3x^{\frac{1}{2}} - \dfrac{9}{4}x^{-\frac{3}{2}}$ | M1 | Attempts to differentiate. Award for one power decreasing by one, indices must be processed. Cannot award if just differentiating top and bottom of fraction separately. |
| $\dfrac{dy}{dx} = 3x^{\frac{1}{2}} - \dfrac{9}{4}x^{-\frac{3}{2}}$ | A1 | May be left unsimplified but indices must be processed |
| $\dfrac{dy}{dx} = 0 \Rightarrow x^2 = \dfrac{3}{4} \Rightarrow x = \dfrac{\sqrt{3}}{2}$ | M1 | Sets $\dfrac{dy}{dx} = 0$ and proceeds to $x^{\pm 2} = \ldots$ or $x^{\pm 4} = \ldots$ following a derivative of form $Ax^{\frac{1}{2}} - Bx^{-\frac{3}{2}},\ A,B > 0$ |
| $x = \dfrac{\sqrt{3}}{2}$ | A1 | Or exact equivalent cso. A correct exact answer can imply final M1A1. Rounded answer with no working (e.g. awrt 0.87) is M0A0. Withhold final mark if $-\dfrac{\sqrt{3}}{2}$ is not rejected. |
9. In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.
A curve has equation
$$y = \frac { 4 x ^ { 2 } + 9 } { 2 \sqrt { x } } \quad x > 0$$
Find the $x$ coordinate of the point on the curve at which $\frac { \mathrm { d } y } { \mathrm {~d} x } = 0$\\
\hfill \mbox{\textit{Edexcel P1 2020 Q9 [6]}}