= Solution
Differentiate the <log-likelihood> again. For $d>0$, the positive interior <maximum-likelihood estimate> gives
$$
\boxed{\ell''(\widehat\theta)=-\frac{d}{\widehat\theta^2}.}
$$
The <observed information> is $-\ell''(\widehat\theta)=d/\widehat\theta^2$, giving the usual local <standard error> approximation $\widehat\theta/\sqrt d$. For $d=0$ there is no positive interior maximum, so the displayed substitution and this <standard error> approximation do not apply; at every positive $\theta$ the second derivative is zero.
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