5 A model car moves so that its distance, \(x\) centimetres, from a fixed point \(O\) after time \(t\) seconds is given by
$$x = \frac { 1 } { 2 } t ^ { 4 } - 20 t ^ { 2 } + 66 t , \quad 0 \leqslant t \leqslant 4$$
- Find:
- \(\frac { \mathrm { d } x } { \mathrm {~d} t }\);
- \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } }\).
- Verify that \(x\) has a stationary value when \(t = 3\), and determine whether this stationary value is a maximum value or a minimum value.
- Find the rate of change of \(x\) with respect to \(t\) when \(t = 1\).
- Determine whether the distance of the car from \(O\) is increasing or decreasing at the instant when \(t = 2\).