7.Figure 2 shows a rectangular section of marshland,$O A B C$ ,which is $a$ metres long by $b$ metres wide,where $a > b$ .

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{175528b0-6cd1-4d0d-a6b3-28ac980f74f3-22_360_847_340_609}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}

Edgar intends to get from $O$ to $B$ in the shortest possible time.In order to do this,he runs along edge $O A$ for a distance $x$ metres $( 0 \leqslant x < a )$ to the point $D$ before wading through the marsh directly from $D$ to $B$ .

Edgar can wade through the marsh at a constant speed of $1 \mathrm {~ms} ^ { - 1 }$ ,and he can run along the edge of the marsh at a constant speed of $\lambda \mathrm { ms } ^ { - 1 }$ ,where $\lambda > 1$
\begin{enumerate}[label=(\alph*)]
\item By finding an expression in terms of $x$ for the time taken,$t$ seconds,for Edgar to reach $B$ from $O$ ,show that

$$\frac { \mathrm { d } t } { \mathrm {~d} x } = \frac { 1 } { \lambda } - \frac { a - x } { \sqrt { ( a - x ) ^ { 2 } + b ^ { 2 } } }$$
\item \begin{enumerate}[label=(\roman*)]
\item Find,in terms of $a , b$ and $\lambda$ ,the value of $x$ for which $\frac { \mathrm { d } t } { \mathrm {~d} x } = 0$
\item Show that this value of $x$ lies in the interval $0 \leqslant x < a$ provided $\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }$
\item For $\lambda$ in this range,show that the value of $x$ found in(b)(i)gives a minimum value of $t$ .
\end{enumerate}\item Find the minimum time taken for Edgar to get from $O$ to $B$ if
\begin{enumerate}[label=(\roman*)]
\item $\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }$
\item $1 < \lambda < \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }$

Edgar's friend,Frankie,also runs at a constant speed of $\lambda \mathrm { m } \mathrm { s } ^ { - 1 }$ .Frankie runs along the edges $O A$ and $A B$ .Given that $\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }$
\end{enumerate}\item find the range of values of $\lambda$ for which Frankie gets to $B$ from $O$ in a shorter time than Edgar's minimum time.
\end{enumerate}

\hfill \mbox{\textit{Edexcel AEA 2019 Q7 [22]}}