= Solution
Write $s=\sum_i\log x_i<0$. The <score function> is
$$
\ell'(\theta)=\frac{N}{\theta+1}+s.
$$
Setting it to zero gives the <maximum-likelihood estimator>
$$
\boxed{\widehat\theta=-\frac{N}{\sum_i\log x_i}-1}.
$$
This lies above $-1$. The <observed Fisher information> is
$$
\boxed{J(\theta)=-\ell''(\theta)=\frac{N}{(\theta+1)^2}>0}.
$$
Consequently the <log-likelihood> is strictly concave throughout its domain. Its stationary point is the unique global maximum, not merely a candidate local maximum. Indeed the <log-likelihood> tends to minus infinity as either $\theta\downarrow-1$ or $\theta\to\infty$.
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