5
\includegraphics[max width=\textwidth, alt={}, center]{1fa404d4-5e14-4356-9b6d-f176d5a9f6db-08_773_1161_278_443}
The diagram shows part of the curve \(\mathrm { y } = \mathrm { xsech } ^ { 2 } \mathrm { x }\) and its maximum point \(M\).
- Show that, at \(M\),
$$2 x \tanh x - 1 = 0$$
and verify that this equation has a root between 0.7 and 0.8 .
- By considering a suitable set of rectangles, use the diagram to show that
\(\sum _ { r = 2 } ^ { n } r \operatorname { sech } ^ { 2 } r < n \tanh n + \operatorname { lnsechn } - \tanh 1 - \operatorname { lnsech } 1\).