Derive logistic regression objective and gradients

Consider binary logistic regression with data $\{ (x_i, y_i) \}_{i=1}^m$, where $y_i \in \{0,1\}$. Using the sigmoid $\sigma(z)=1/(1+e^{-z})$ and linear score $z_i = w^T x_i + b$: a) Write the exact maximum-likelihood optimization objective, and state clearly whether it is a maximization or minimization, both without and with L2 regularization of strength $\lambda \ge 0$. b) Write out the explicit negative log-likelihood (cross-entropy) $L(w,b)$. c) Derive the gradients $\partial L/\partial w$ and $\partial L/\partial b$. d) Clarify the distinction among 'objective', 'loss', and 'regularized objective' in this context. Overview: This question tests understanding of logistic regression, maximum likelihood estimation, cross-entropy (negative log-likelihood), L2 regularization, and the ability to compute model gradients for binary classification.