The curve C has equation y = e^(2x) sin x.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. Part (a): M1 for attempting product rule, (u=e^(2x), v=sin x) to get e^(2x)cos x + 2e^(2x)sin x
2. Part (a): A1 for e^(2x)cos x + 2e^(2x)sin x
3. Part (a): A1 for factorising to e^(2x)(2sin x + cos x) (Shown)
4. Part (b): M1 for attempting product rule on dy/dx, u=e^(2x), v=(2sin x + cos x). Correct form: u'v + uv'
5. Part (b): A1 for 2e^(2x)(2sin x + cos x) + e^(2x)(2cos x - sin x)
6. Part (b): M1 for factorising out e^(2x)
7. Part (b): A1 for d^2y/dx^2 = e^(2x)(3sin x + 4cos x)
8. Part (c): M1 for setting dy/dx = 0, leading to 2sin x + cos x = 0 or tan x = -1/2
9. Part (c): A1 for x = arctan(-1/2) (e.g., -0.4636 or 2.6779) and x = arctan(-1/2) + π (or equivalent general solution using tan(x+nπ) = tan x)
10. Part (c): A1 for listing two exact values, e.g., x = arctan(-1/2) and x = arctan(-1/2) + π (or any two exact values separated by π)