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] \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 \(\lambda > 1\)
（a）By finding an expression in terms of \(x\) for the time taken，\(t\) seconds，for Edgar to reach \(B\) from \(O\) ，show that $$\frac { \mathrm { d } t } { \mathrm {~d} x } = \frac { 1 } { \lambda } - \frac { a - x } { \sqrt { ( a - x ) ^ { 2 } + b ^ { 2 } } }$$ （b）（i）Find，in terms of \(a , b\) and \(\lambda\) ，the value of \(x\) for which \(\frac { \mathrm { d } t } { \mathrm {~d} x } = 0\)
（ii）Show that this value of \(x\) lies in the interval \(0 \leqslant x < a\) provided \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
（iii）For \(\lambda\) in this range，show that the value of \(x\) found in（b）（i）gives a minimum value of \(t\) ．
（c）Find the minimum time taken for Edgar to get from \(O\) to \(B\) if
（i）\(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
（ii） \(1 < \lambda < \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\) 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 \(\lambda \geqslant \sqrt { 1 + \frac { b ^ { 2 } } { a ^ { 2 } } }\)
（d）find the range of values of \(\lambda\) for which Frankie gets to \(B\) from \(O\) in a shorter time than Edgar＇s minimum time．