 Solution

Independence and the Poisson probabilities give the log-likelihoodIts score and curvature areIf some $x_i\ne 0$, this is strictly concave. A finite maximum is therefore the unique zero of the score when such a zero exists. Newton iteration givesStart from a finite value, compute the score and curvature, and iterate until both the score and step are small. A line search that shortens a step until the likelihood increases improves robustness. Strict concavity identifies any converged stationary point as the global maximum. A finite root need not exist, for example when all observations are zero and all $x_i>0$, in which case the likelihood supremum is at $\beta \to -\infty$. If all $x_i=0$, the parameter is unidentifiable.