June 2024 Paper 2 Q8
8 The equation of a curve is
\(y = 2x^3 + 3mx^2 - 9mx + 4\).
Determine the range of values of \(m\) for which the curve has no stationary values. [6]
| Scheme | Marks | AO |
|---|---|---|
| \(\frac{\mathrm{d}y}{\mathrm{d}x} = 6x^2 + 6mx - 9m\) | M1* A1 | 3.1a 1.1 |
| \((6m)^2 - 4 \times 6 \times (-9m)\) oe seen | M1dep* | 3.1a |
| two values of \(m\) obtained from their discriminant \(36m^2 + 216m \lt 0\) or \(36m^2 + 216m = 0\) oe | M1 | 2.1 |
| 0 and \(-6\) identified | A1 | 1.1 |
| \(-6 \lt m \lt 0\) oe | A1 | 3.2a |
| [6] |
Notes
M1*: differentiation of all 4 terms with 3 of the 4 terms differentiated correctly
A1: all correct
M1dep*: discriminant for their \(6, 6m\) and \(-9m\);
may see \((2m)^2 - 4 \times 2 \times (-3m)\)
M1: dependent on obtaining discriminant from their derivative;
M0 for use of their discriminant \(\gt 0\) or \(\geqslant 0\)
NB \(4m^2 + 24m \lt 0\) or \(4m^2 + 24m = 0\)
A1: inequality or interval must be strict