Starting from the result that
\[
\.h(t) >0\ \mathrm{for}\ 0< t < x \Longrightarrow \int_0^x \.h(t)\ud t
> 0 \,,
\]
show that, if \(\.f''(t) > 0\) for \(0 < t < x_0\) and \(\.f(0)=\.f'(0) =0\),
then \(\.f(t)>0\) for \(0 < t < x_0\).
Show that,
for \(0 < x < \frac12\pi\),
\[
\cos x \cosh x <1
\,.
\]
Show that, for \(0 < x < \frac12\pi\),
\[
\frac 1 {\cosh x} < \frac {\sin x} x < \frac x {\sinh x} \,.
\]
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Show that, for \(0 < x < \frac12\pi\), \(\tanh x < \tan x\).