Prove, using exponential functions, that
$$\sinh 2 x = 2 \sinh x \cosh x$$
Differentiate this result to obtain a formula for \(\cosh 2 x\).
Sketch the curve with equation \(y = \cosh x - 1\).
The region bounded by this curve, the \(x\)-axis, and the line \(x = 2\) is rotated through \(2 \pi\) radians about the \(x\)-axis. Find, correct to 3 decimal places, the volume generated. (You must show your working; numerical integration by calculator will receive no credit.)
Show that the curve with equation
$$y = \cosh 2 x + \sinh x$$
has exactly one stationary point.
Determine, in exact logarithmic form, the \(x\)-coordinate of the stationary point.