June 2025 Paper 2 Q6
6 The equation of a curve is
\[y = x^4 - 8x^3 + 20.\](a) Show that the gradient function, \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\), is \[\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4x^2(x - 6).\] [2]
(b) 
Axes printed in the Printed Answer Booklet
(i) On the axes in the Printed Answer Booklet, sketch the gradient function of \(y\). [2]
(ii) State the coordinates of the intercepts of the gradient function of \(y\). [1]
(c) Hence state the range of values of \(x\) for which \(y = x^4 - 8x^3 + 20\) is an increasing function. Express your answer in set notation. [1]
| Scheme | Marks | AO |
|---|---|---|
| \(4x^3 - 24x^2\) | M1 | 2.1 |
| \(= 4x^2(x - 6)\) | A1 | 1.1 |
| [2] |
Notes
M1: allow one coefficient error or inclusion of +20 but not both
A1: AG
| Scheme | Marks | AO |
|---|---|---|
(i) | B1 B1 | 1.1 1.1 |
| [2] | ||
| (ii) (0,0) and (6,0) | B1 | 1.1 |
| [1] |
Notes
(i) B1: cubic of correct orientation and shape
(i) B1: cubic of correct orientation with maximum at (0,0) with positive \(x\)-intercept
(ii) B1: B0 if any additional values seen
| Scheme | Marks | AO |
|---|---|---|
| \(\{x : x \gt 6\}\) or \(\{x : x \geqslant 6\}\) | B1 | 2.2a |
| [1] |
Notes
Allow \((6, \infty)\) or \([6, \infty)\)