Show that if the log-likelihood L( I x) is a concave function of for each scalar

Show that if the log-likelihood L(θ I x) is a concave function of θ for each scalar x (that is, L"(θ l x) ≤ ( 0 for all θ), then the likelihood function L(θ l x) fore given an n-sample x = (x₁, x₂, ... , x_n) has a unique maximum. Prove that this is the case if the observations x_i come from a logistic density

where θ is an unknown real parameter. Fill in the details of the Newton-Raphson method and the method of scoring for finding the position of the maximum, and suggest a suitable starting point for the algorithms.

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p(x|0) = exp(0 − x)/{1+ exp(0 - x)}² (-∞ < x < ∞),