June 2018 Paper 2 Q21
21 The continuous curve \(\ y = \mathrm{f}(x)\ \) has exactly two stationary points.
Here is some information about the curve.
| \(x \lt -1\) | \(x = -1\) | \(-1 \lt x \lt 2\) | \(x = 2\) | \(x \gt 2\) |
|---|---|---|---|---|
| \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) is positive | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) is zero | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) is negative | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) is zero | \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) is negative |
\(\mathrm{f}(-1) = 3 \quad\) and \(\quad \mathrm{f}(2) = 1\)
State the coordinates and the nature of each of the stationary points. [3 marks]
| Answer | Mark | Comments |
|---|---|---|
| \((-1,\ 3)\) or \((2,\ 1)\) | M1 | |
| \((-1,\ 3)\) max(imum) | A1 | |
| \((2,\ 1)\) (point of) inflection | A1 | SC1 \((3,\ -1)\) max(imum) and \((1,\ 2)\) (point of) inflection |
Additional guidance
| One correct point and nature | M1A1 |
| Ignore reference to ‘stationary points’ or ‘turning points’ or ‘local’ | |
| Condone p o i for point of inflection |