Prove that minimal of function below is attainable

Question

Prove that real value function :

$f(x)=-\sum_{i=1}^{n}\log(b_i-a_i^Tx)$ with domain $\{x:Ax <b\}$ where $a_i \in \mathbb{R}^m$ is row of $A$ .

1.Can attain it's minimum when $f$ is bounded below.

2.optimal set is affine that is $\{x^\star +v:Av = 0\}$ where $x^\star$ is the minimal value

You can use the result that :

1. domain of f is unbounded iff there is a $v \ne 0$ such that $Av \le 0$
2. f is unbounded below iff there is a $v$ with $Av \le 0$ and $Av \ne 0$