Consider a binary classification problem where you are trying to classify whether a tumor is benign $(0)$ or malignant $(1)$ based on a single feature xinR.

$($a$)$ Define the logistic regression model for this problem by choosing appropriate numbers of parameters, i$.$e$.$ find a probability function of a test feature $x$ parametrized by the parameters, which is call a "hypothesis".

$($b$)$ Write the cost function $l$ with respect to the parameters.

$($c$)$ Derive the expression for the gradient of $l$ with respect to the parameters.