The cross entropy is a convex function of the weights \( \boldsymbol{\theta} \) and, therefore, any local minimizer is a global minimizer.
Minimizing this cost function with respect to the two parameters \( \theta_0 \) and \( \theta_1 \) we obtain
$$ \frac{\partial \mathcal{C}(\boldsymbol{\theta})}{\partial \theta_0} = -\sum_{i=1}^n \left(y_i -\frac{\exp{(\theta_0+\theta_1x_i)}}{1+\exp{(\theta_0+\theta_1x_i)}}\right), $$and
$$ \frac{\partial \mathcal{C}(\boldsymbol{\theta})}{\partial \theta_1} = -\sum_{i=1}^n \left(y_ix_i -x_i\frac{\exp{(\theta_0+\theta_1x_i)}}{1+\exp{(\theta_0+\theta_1x_i)}}\right). $$