Standard +0.8 This question requires finding first and second derivatives of terms with negative/fractional powers, setting dy/dx = 0, then evaluating dΒ²y/dxΒ² at the turning point. While the algebraic manipulation is standard A-level fare, the 7-mark allocation and requirement to 'fully justify' suggests students must show the second derivative is definitively positive (using the fact that a, b, c are positive constants and the turning point condition). This goes beyond routine differentiation into careful algebraic reasoning, placing it moderately above average difficulty.
A curve has equation
$$y = \frac{a}{\sqrt{x}} + bx^2 + \frac{c}{x^3} \quad \text{for } x > 0$$
where \(a\), \(b\) and \(c\) are positive constants.
The curve has a single turning point.
Use the second derivative of \(y\) to determine the nature of this turning point.
You do not need to find the coordinates of the turning point.
Fully justify your answer.
[7 marks]
Question 9:
9 | Expresses all terms as powers
of x at least 2 correct. PI | 1.1a | M1 | y = + bx2 +
β0.5 β3
πππ₯π₯ πππ₯π₯
= - + 2bx β 3
dπ¦π¦ 1 β1.5 β4
2πππ₯π₯ πππ₯π₯
dπ₯π₯
= + 2b + 12
2
d π¦π¦ 3 β2.5 β5
2 4πππ₯π₯ πππ₯π₯
Asd aπ₯π₯, b, c and x are all > 0, all terms
must be positive
so is positive
2
d π¦π¦
2
so turnindgπ₯π₯ point is a minimum
Differentiates at least one of
their negative powers correctly | 1.1a | M1
Obtains completely correct
differential | 1.1b | A1
Differentiates again, powers and
signs correct | 1.1a | M1
Deduces that is positive
2
d π¦π¦ | 2.2a | A1F
2
dπ₯π₯
Explains that positive second
differential shows that turning
point is a minimum | 2.4 | E1F
Shows completely correct
mathematics throughout,
including coefficients of
2
d π¦π¦
must refer to a, b, c and x being
2
> 0 dπ₯π₯ | 2.1 | R1
Total | 7
Q | Marking instructions | AO | Marks | Typical solution
A curve has equation
$$y = \frac{a}{\sqrt{x}} + bx^2 + \frac{c}{x^3} \quad \text{for } x > 0$$
where $a$, $b$ and $c$ are positive constants.
The curve has a single turning point.
Use the second derivative of $y$ to determine the nature of this turning point.
You do not need to find the coordinates of the turning point.
Fully justify your answer.
[7 marks]
\hfill \mbox{\textit{AQA AS Paper 1 2021 Q9 [7]}}