= Solution
Assume the <scale family> has the differentiability needed for <Fisher information>, and the following constant is finite and positive. With $\ell_f(u)=\log f(u)$, the <score function> is
$$
\partial_\sigma\log p(y\mid\sigma)
=-\frac{1+u\ell_f'(u)}{\sigma},\qquad u=y/\sigma.
$$
Part (b) therefore gives
$$
I(\sigma)=\frac C{\sigma^2},\qquad
C=\int[1+u\ell_f'(u)]^2f(u)\,du,
$$
where $C$ is independent of $\sigma$. The <Jeffreys prior for a scale parameter> is consequently
$$
\boxed{p_J(\sigma)\propto\sigma^{-1},\qquad\sigma>0.}
$$
The reciprocal-of-a-reciprocal in the TeX is a transcription error; the PDF has $\sigma^{-1}$. The expected-Hessian information identity should not be applied indiscriminately to nonregular parameter-dependent supports.
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