Show that there is no point where the tangent to the curve is horizontal.

Find the coordinates of the points where the tangent to the curve is vertical.

$x\frac{{{\text{d}}y}}{{{\text{d}}x}} + y = 2y\frac{{{\text{d}}y}}{{{\text{d}}x}}$     M1A1

a horizontal tangent occurs if $\frac{{{\text{d}}y}}{{{\text{d}}x}} = 0$ so $y = 0$     M1

we can see from the equation of the curve that this solution is not possible $(0 = 4)$ and so there is not a horizontal tangent     R1

[4 marks]

$\frac{{{\text{d}}y}}{{{\text{d}}x}} = \frac{y}{{2y - x}}$ or equivalent with $\frac{{{\text{d}}x}}{{{\text{d}}y}}$

the tangent is vertical when $2y = x$     M1

substitute into the equation to give $2{y^2} = {y^2} + 4$     M1

$y =  \pm 2$     A1

coordinates are $(4,{\text{ }}2),{\text{ }}( - 4,{\text{ }} - 2)$     A1

[4 marks]

Total [8 marks]