June 2025 Paper 2 Q15
15 For a curve \(\quad y = \mathrm{f}(x)\)
\[\frac{\mathrm{d}y}{\mathrm{d}x} = 1 - \frac{9}{x^2}\](a) Show that the curve has a turning point where \(\quad x = 3\) [1 mark]
(b)
Use \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) to work out the nature of the turning point.
You must show your working. [2 marks]
| Answer | Mark | Comments |
|---|---|---|
| Alternative method 1: equates gradient to 0 | ||
| \(1 - \dfrac{9}{x^2} = 0\) and \(x = 3\) | B1 | accept \(9x^{-2}\) for \(\dfrac{9}{x^2}\) ignore \(x = -3\) |
| Alternative method 2: substitutes \(x = 3\) into gradient | ||
| \(1 - \dfrac{9}{3^2} = 0\) or \(1 - \dfrac{9}{9} = 0\) | B1 | may be seen in stages accept \(9 \times 3^{-2}\) for \(\dfrac{9}{3^2}\) |
Additional guidance
| Alt1 and alt 2 with errors in working is B0 eg Alt 1 \(\ 1 - \dfrac{9}{x^2} = 0 \quad -1 = \dfrac{9}{x^2} \quad x = 3\) | B0 |
| After B1 response ignore non-contradictory statements eg Alt 2 \(\ 1 - \dfrac{9}{3^2} = 0 \quad\) A turning point at (3, 0) | B1 |
| Alt 2 \(\ \dfrac{9}{3^2} = 1 \quad 1 - 1 = 0\) | B1 |
| Alt 2 \(\ \dfrac{9}{3^2} = 1\) with no further correct work | B0 |
| Alt 1 \(\ 1 - \dfrac{9}{x^2} = 0 \quad x = \pm 3\) | B1 |
| Answer | Mark | Comments |
|---|---|---|
| \(18x^{-3}\) with no other terms | M1 | oe eg \(-9 \times -2x^{-2 - 1}\) with no other terms |
| M1 awarded and \(18 \times 3^{-3} \gt 0\) and minimum or M1 awarded and \(\dfrac{18}{27}\) and minimum | A1 | oe eg \(18x^{-3}\) with no other terms and this is positive when \(x\) is 3 and minimum or M1 awarded and \(\dfrac{2}{3}\) and minimum allow [0.66, 0.67] for \(\dfrac{18}{27}\) any evaluations of second derivative must be correct |
Additional guidance
| M1 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 18x^{-3}\) with no other terms is awarded M1 and may be awarded A1 with further correct working | |
| For A1 \(x = 3\) must be used or stated eg \(18x^{-3}\) is positive for all positive values of \(x\) and minimum | M1A0 |
| Only considering signs of \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) either side of \(x = 3\) | M0A0 |
| Accept eg local min or minima for minimum | |
| After differentiating to \(18x^{-3}\) with no other terms, ignore further differentiation eg \(18x^{-3}\) and \(-54x^{-4}\) | M1 |