When the line given by the objective function is parallel to any of the boundary of the feasible region it gives?

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Some basic linear algebra is essential for this. Have you already studied that a bit?

Say you have point $p$ inside your polyhedron of feasible points inside $\mathbb{R}^n$, and let's say you want to maximize the objective function. The objective function is linear, given by some $c \in \mathbb{R}^n$. If $v \in \mathbb{R}^n$ and $c \cdot v = 0$, then $p+v$ has the same objective value as $p$. In other words, one does not improve $p$ by moving it in a direction orthogonal to the objective function.

If $v$ has some component parallel to $c$, then $p+v$ will be either strictly better or strictly worse than $p$. Therefore, if you are the point $p$ and you want to improve your objective value, then you should look for a direction $v$ in which you can move which has some positive component parallel to $c$, that is $c \cdot v > 0$. You may not be able to walk exactly in the direction of $c$ because a face of the polyhedron may stand in the way. But you may be able to walk in a direction having positive $c$ component. The only way you could get stuck is that for every possible improving direction, some face blocks you. In other words for every $v$ in the open halfspace $\{v:c\cdot v > 0\}$, the point $p+v$ lies outside the feasible polyhedron. This can only happen if either you already stand at a vertex, or you stand on a face of optimal solutions parallel to the hyperplane $\{v:c\cdot v = 0\}$. If you do not already stand at a vertex, then while staying on the hyperplane, you can walk to a vertex without changing your objective value.

In the above explanation, I have used the boundedness of the feasible region: Without boundedness, it might be possible to walk in an improving direction forever, or it might be possible to walk along a face of optimal solutions without ever arriving at a vertex. I have also used convexity: If I do not stand at an optimal solution, then there will always exist an improving direction.