Moderate -0.3 This is a straightforward stationary points question requiring standard differentiation of power functions (rewriting surds as fractional powers), setting the derivative to zero, and using the second derivative test. All steps are routine A-level techniques with no conceptual challenges, making it slightly easier than average.
3. The curve \(C\) has equation
$$y = 8 \sqrt { x } + \frac { 18 } { \sqrt { x } } - 20 \quad x > 0$$
a. Find
i) \(\frac { d y } { d x }\)
ii) \(\frac { d ^ { 2 } y } { d x ^ { 2 } }\)
b. Use calculus to find the coordinates of the stationary point of \(C\).
c. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.
Achieves a correct \(\frac{d^2y}{dx^2}\) for their \(\frac{dy}{dx}\)
Part b:
Answer
Marks
Guidance
\(\left(\frac{9}{4}, 4\right)\)
A1
Correct answer only
M1
Rearrange to find \(x\), correct answer only
M1
Substitutes their \(x\) value(s) into the given equation
Part c:
Answer
Marks
Guidance
\(\frac{d^2y}{dx^2} > 0\), so is a minimum
A1
Fully correct solution including a correct numerical second derivative (awrt) and a reference to positive or \(> 0\) and a conclusion
M1
Substitutes their \(x\) value(s) into their second derivative
### Part a.i:
$\frac{dy}{dx} = 4x^{-\frac{1}{2}} - 9x^{-\frac{3}{2}}$ | A1 | Achieves a correct $\frac{dy}{dx}$
| M1 | Differentiates to $\frac{dy}{dx} = Ax^{-\frac{1}{2}} + Bx^{-\frac{3}{2}}$
### Part a.ii:
$\frac{d^2y}{dx^2} = -2x^{-\frac{3}{2}} + \frac{27}{2}x^{-\frac{5}{2}}$ | B1 | Achieves a correct $\frac{d^2y}{dx^2}$ for their $\frac{dy}{dx}$
### Part b:
$\left(\frac{9}{4}, 4\right)$ | A1 | Correct answer only
| M1 | Rearrange to find $x$, correct answer only
| M1 | Substitutes their $x$ value(s) into the given equation
### Part c:
$\frac{d^2y}{dx^2} > 0$, so is a minimum | A1 | Fully correct solution including a correct numerical second derivative (awrt) and a reference to positive or $> 0$ and a conclusion
| M1 | Substitutes their $x$ value(s) into their second derivative
3. The curve $C$ has equation
$$y = 8 \sqrt { x } + \frac { 18 } { \sqrt { x } } - 20 \quad x > 0$$
a. Find\\
i) $\frac { d y } { d x }$\\
ii) $\frac { d ^ { 2 } y } { d x ^ { 2 } }$\\
b. Use calculus to find the coordinates of the stationary point of $C$.\\
c. Determine whether the stationary point is a maximum or minimum, giving a reason for your answer.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q3 [9]}}