OCR
Further Pure Core 1
2018
September
— Question 8
Exam Board
OCR
Module
Further Pure Core 1 (Further Pure Core 1)
Year
2018
Session
September
Topic
Hyperbolic functions
8
Using the definitions of \(\cosh x\) and \(\sinh x\) in terms of \(\mathrm { e } ^ { x }\) and \(\mathrm { e } ^ { - x }\), show that \(\sinh 2 x = 2 \sinh x \cosh x\).
You are given the function \(\mathrm { f } ( x ) = a \cosh x - \cosh 2 x\), where \(a\) is a positive constant.
Verify that, for any value of \(a\), the curve \(y = \mathrm { f } ( x )\) has a stationary point on the \(y\)-axis.
Find the coordinates of the stationary point found in part (ii).
Determine the maximum value of \(a\) for which the stationary point found in part (ii) is the only stationary point on the curve \(y = \mathrm { f } ( x )\).
You are given that for any value of \(a\) greater than the value found in part (iv) there are three stationary points, the one found in part (ii) and two others, one of which satisfies \(x > 0\).
Find the coordinates of this point when \(a = 6\).
Give your answer in the form \(\left( \cosh ^ { - 1 } p , q \right)\).