June 2023 Paper 2 Q17
17 The function h is given by \(\quad \mathrm{h}(x) = ax(3x^2 - 2) + 5x \quad\) where \(a\) is a positive constant.
h is an increasing function for all values of \(x\).
Work out the possible values of \(a\).
Give your answer as an inequality. [4 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(3ax^3 - 2ax\) | M1 | oe with brackets expanded may be seen in an inequality or equation |
| \(9ax^2 - 2a + 5\) | M1 | oe eg \(3 \times 3ax^{3 - 1} - 2ax^0 + 5x^0\) differentiates their expression derivative must contain 3 terms may be seen in an inequality or equation |
| \(9ax^2 - 2a + 5 \gt 0\) or \(9ax^2 - 2a + 5 \geqslant 0\) | M1dep | oe eg \(9ax^2 - 2a \gt -5\) must be correct implied by \(-2a + 5 \gt 0\) or \(-2a + 5 \geqslant 0\) with no incorrect working dep on M2 |
| \((0 \lt)\; a \lt \dfrac{5}{2}\) or \((0 \lt)\; a \leqslant \dfrac{5}{2}\) | A1 | oe eg \((0 \lt)\; a \lt 2\dfrac{1}{2}\) or \((0 \lt)\; a \lt 2.5\) or \((0 \lt)\; a \leqslant 2\dfrac{1}{2}\) or \((0 \lt)\; a \leqslant 2.5\) |
Additional guidance
| Up to M3 may be awarded for correct work with no answer or incorrect answer, even if this is seen amongst multiple attempts | |
| Condone incorrect notation eg \(\mathrm{h} = 9ax^2 - 2a + 5\) | M1M1 |
| Working with = 0 does not score 3rd M unless recovered to the correct answer eg \(3ax^3 - 2ax + 5x \qquad 9ax^2 - 2a + 5 = 0 \qquad a = 2.5\) Answer \(a \lt 2.5\) | M1M1 M1A1 |
| Working with \(\lt 0\) (or \(\leqslant 0\)) does not score 3rd M and cannot be recovered eg \(3ax^3 - 2ax + 5x \qquad 9ax^2 - 2a + 5 \lt 0\) \(-2a + 5 \lt 0 \qquad\) Answer \(a \lt 2.5\) | M1M1M0 A0 |
| Correct answer may come from other incorrect methods eg \(3ax^3 - 2ax + 5x \qquad -2ax + 5x \gt 0\) \(-2a + 5 \gt 0 \qquad\) Answer \(a \lt \dfrac{5}{2}\) | M1M0 M0A0 |
| Allow the first two marks for fully correct use of the product rule for differentiation (M0M0 if not fully correct) eg \(ax \times 6x + (3x^2 - 2) \times a + 5\) | M1M1 |
| \(3ax^3 - 2ax + 5x\) \(9x^2 - 2a + 5\) | M1 M0 |
| \(3ax^2 - 2ax + 5x\) \(6ax - 2a + 5\) | M0 M1 |
| \(3ax^3 - 2ax\) \(9ax^2 - 2a\) (their derivative does not contain 3 terms) | M1 M0 |
| Ignore higher derivatives | |
| Allow interval notation eg (0, 2.5] | |
| Condone \(0 \leqslant a \lt 2.5\) or \(0 \leqslant a \leqslant 2.5\) |