Given that \(y = \tanh ^ { - 1 } x\), for \(- 1 < x < 1\), prove that \(y = \frac { 1 } { 2 } \ln \left( \frac { 1 + x } { 1 - x } \right)\).
It is given that \(\mathrm { f } ( x ) = a \cosh x - b \sinh x\), where \(a\) and \(b\) are positive constants.
(a) Given that \(b \geqslant a\), show that the curve with equation \(y = \mathrm { f } ( x )\) has no stationary points.
(b) In the case where \(a > 1\) and \(b = 1\), show that \(\mathrm { f } ( x )\) has a minimum value of \(\sqrt { a ^ { 2 } - 1 }\).