Show that the curve with equation
$$y = 3 \sinh x - 2 \cosh x$$
has no turning points.
Show that the curve crosses the \(x\)-axis at \(x = \frac { 1 } { 2 } \ln 5\). Show that this is also the point at which the gradient of the curve has a stationary value.
Sketch the curve.
Express \(( 3 \sinh x - 2 \cosh x ) ^ { 2 }\) in terms of \(\sinh 2 x\) and \(\cosh 2 x\).
Hence or otherwise, show that the volume of the solid of revolution formed by rotating the region bounded by the curve and the axes through \(360 ^ { \circ }\) about the \(x\)-axis is
$$\pi \left( 3 - \frac { 5 } { 4 } \ln 5 \right) .$$
Option 2: Investigation of curves
\section*{This question requires the use of a graphical calculator.}