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]