Given that \(y = \sinh x\), use the definition of \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\) to show that \(x = \ln \left( y + \sqrt { y ^ { 2 } + 1 } \right)\).
A curve has equation \(y = 6 \cosh ^ { 2 } x + 5 \sinh x\).
Show that the curve has a single stationary point and find its \(x\)-coordinate, giving your answer in the form \(\ln p\), where \(p\) is a rational number.
The curve lies entirely above the \(x\)-axis. The region bounded by the curve, the coordinate axes and the line \(x = \cosh ^ { - 1 } 2\) has area \(A\).
Show that
$$A = a \cosh ^ { - 1 } 2 + b \sqrt { 3 } + c$$
where \(a\), \(b\) and \(c\) are integers. [0pt]
[5 marks]
\(7 \quad\) Given that \(p \geqslant - 1\), prove by induction that, for all integers \(n \geqslant 1\),
$$( 1 + p ) ^ { n } \geqslant 1 + n p$$
[6 marks]