Find stationary points of hyperbolic curves

A question is this type if and only if it asks to find and determine the nature of stationary points (or turning points) on a curve defined by hyperbolic functions using differentiation.

28 questions · Standard +0.9

4.07d Differentiate/integrate: hyperbolic functions
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SPS SPS FM 2021 November Q7
7 marks Challenging +1.3
The curve with equation $$y = -x + \tanh(36x), \quad x \geq 0,$$ has a maximum turning point \(A\).
  1. Find, in exact logarithmic form, the \(x\)-coordinate of \(A\). [4 marks]
  2. Show that the \(y\)-coordinate of \(A\) is $$\frac{\sqrt{35}}{6} - \frac{1}{36}\ln(6 + \sqrt{35}).$$ [3 marks]
SPS SPS FM Pure 2025 February Q3
7 marks Challenging +1.2
The curve \(C\) has equation $$y = 31\sinh x - 2\sinh 2x \quad x \in \mathbb{R}$$ Determine, in terms of natural logarithms, the exact \(x\) coordinates of the stationary points of \(C\). [7]
OCR Further Pure Core 2 2018 March Q6
12 marks Challenging +1.2
In this question you must show detailed reasoning.
  1. Find the coordinates of all stationary points on the graph of \(y = 6\sinh^2 x - 13\cosh x\), giving your answers in an exact, simplified form. [9]
  2. By finding the second derivative, classify the stationary points found in part (i). [3]