5 A curve \(C\) has parametric equations
$$x = \cos t , \quad y = 3 + 2 \cos 2 t , \quad \text { where } 0 \leqslant t \leqslant \pi$$
- Express \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) in terms of \(t\) and hence show that the gradient at any point on \(C\) cannot exceed 8 .
- Show that all points on \(C\) satisfy the cartesian equation \(y = 4 x ^ { 2 } + 1\).
- Sketch the curve \(y = 4 x ^ { 2 } + 1\) and indicate on your sketch the part which represents \(C\).