June 2022 Paper 1 Q20
20 The curve \(\quad y = x^4 - 18x^2 \quad\) has three stationary points.
Work out the coordinates of the three stationary points and determine their nature.
You must show your working. [6 marks]
| Answer | Mark | Comments |
|---|---|---|
| \(\left(\dfrac{\mathrm{d}y}{\mathrm{d}x} =\right) 4x^3 - 36x\) | M1 | either term correct |
| their \(\dfrac{\mathrm{d}y}{\mathrm{d}x} = 0\) | M1dep | could be written as \(x(x^2 - 9) = 0\) or \(4x(x^2 - 9) = 0\) follow through an incorrect differentiation as long as it has at least one term correct |
| \(4x(x + 3)(x - 3)\;\;(= 0)\) | M1dep | oe eg \(x(x + 3)(x - 3)\;\;(= 0)\) solutions could be gained by using the factor theorem |
| \((-3, -81) \quad (0, 0) \quad (3, -81)\) | A1 | may be seen in calculation rather than put in coordinates at this stage |
| \(\left(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2} =\right) 12x^2 - 36\) and when \(x = -3\) \(\left(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\right)\) = 72 and/or positive or when \(x = 0\) \(\left(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\right)\) \(= -36\) and/or negative or when \(x = 3\) \(\left(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\right)\) = 72 and/or positive or any check to both sides of one of their solutions to give one side with a negative gradient and one side with a positive gradient | M1dep | dependent on M3 oe correct y coordinates not required for this M mark any one point assessed correctly (don’t need to state max or min at this stage) but if the value of f’’(x) is worked out incorrectly then penalise. The value of f’’(x) may not be shown and then the correct statement will suffice. eg \(x = -4 \quad \dfrac{\mathrm{d}y}{\mathrm{d}x} \lt 0\) \(x = -1 \quad \dfrac{\mathrm{d}y}{\mathrm{d}x} \gt 0\) \(x = 1 \quad \dfrac{\mathrm{d}y}{\mathrm{d}x} \lt 0\) \(x = 4 \quad \dfrac{\mathrm{d}y}{\mathrm{d}x} \gt 0\) |
| \((-3, -81)\) Minimum \((0, 0)\) Maximum \((3, -81)\) Minimum | A1 | all three points must have been determined correctly to gain this mark this could imply the previous mark by use of a correct sketch graph or a statement that says a positive quartic has these stationary points |
Additional guidance
| Condone incorrect writing of \(\dfrac{\mathrm{d}y}{\mathrm{d}x}\) and \(\dfrac{\mathrm{d}^2y}{\mathrm{d}x^2}\) even if it’s just \(y =\) as long as it’s recovered to get the correct nature of the turning points |