4.07d Differentiate/integrate: hyperbolic functions

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]

Given that $$y = \cos x \sinh x \quad x \in \mathbb{R}$$

1. show that $$\frac{d^4y}{dx^4} = ky$$ where \(k\) is a constant to be determined. [5]
2. Hence determine the first three non-zero terms of the Maclaurin series for \(y\), giving each coefficient in simplest form. [3]

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]