The coordinates $( x , y )$ of a moving point P are given by the following functions in time $t$:
$$\begin{aligned} & x = 4 t - \sin 4 t \\ & y = 4 - \cos 4 t \end{aligned}$$
(1) The derivatives of $x$ and $y$ with respect to $t$ are
$$\begin{aligned} \frac { d x } { d t } & = \mathbf { A } ( \mathbf { A } - \mathbf { B } \cos 4 t ) \\ \frac { d y } { d t } & = \mathbf { C } \sin 4 t . \end{aligned}$$
Hence we have
$$\left( \frac { d x } { d t } \right) ^ { 2 } + \left( \frac { d y } { d t } \right) ^ { 2 } = \mathbf { D E } \sin ^ { 2 } \mathbf { F } t$$
(2) As the point P moves from the time $t = 0$ to the time $t = 2 \pi$, its speed $v$ is maximized a total of $\mathbf { G }$ times. Let us denote by $t _ { 0 }$ the moment of the first time the speed is maximized and the moment of the last time it is maximized by $t _ { 1 }$. Then
$$t _ { 0 } = \frac { \mathbf { H } } { \mathbf { I } } \pi , \quad t _ { 1 } = \frac { \mathbf { J } } { \mathbf { I } } \pi$$
and the maximum speed is $v = \mathbf { L }$.
(3) For $t _ { 0 }$ and $t _ { 1 }$ in (2), the distance that point P moves during the period from $t = t _ { 0 }$ to $t = t _ { 1 }$ is $\mathbf{MN}$.