Let $X_1, \ldots, X_n$ denote a sample from a population with density $\theta$ when $f(x, \theta) = c \theta^c x^{-(c + 1)}$, $x \geq \theta$; $c$ constant > 0; $\theta > 0$.
I came up with a solution to this problem, but it seems a bit iffy so I thought I'd ask the community.
The likelihood of this sample is \begin{align} L_{X_1, \ldots, X_n}(\theta) & = c^n \theta^{nc} \left(\prod_{i=1}^n X_i \right)^{-(c + 1)} \prod_{i=1}^n 1(X_i \geq \theta), \\ & = c^n \theta^{nc} \left(\prod_{i=1}^n X_i \right)^{-(c + 1)} 1(X_{(1)} \geq \theta). \end{align}
For $X_{(1)} < 0$ the likelihood is 0. For $X_{(1)} \geq 0$, the likelihood is proportional to $\theta^{nc}$ and it is increasing in $\theta$, but the maximum value that $\theta$ can attain on $(0, X_{(1)}]$ is at the right boundary point. Hence the MLE of $\theta$ is $X_{(1)}$.
Is this correct? If yes, is there a more principled way of solving this type of problems that don't require analyzing the function itself? Furthermore, if the constraint was of type $X_i > \theta$, then the MLE wouldn't exist?