Of all the points of the feasible region for maximum or minimum of objective function the points

The correct option is Vertex point of the boundary of the feasible region.

Step-by-step Explanation:

1. Feasible Region in Linear Programming: In a linear programming problem (LPP), the set of all points that satisfy a given system of linear constraints (inequalities or equations) is called the feasible region. This region is typically a convex polygon (in two dimensions) or a convex polyhedron (in higher dimensions).

2. The Corner Point Theorem (or Fundamental Theorem of Linear Programming): According to this mathematical theorem, if an optimal value (either maximum or minimum) of a linear objective function exists, it must occur at one of the corner points, also known as the vertices, of the boundary of the feasible region.

3. Why at the Vertices? The objective function is represented as a linear function, typically of the form:
$$Z=ax+by$$
As we change the value of $Z$, the equation represents a family of parallel straight lines. The optimal value of $Z$ represents the extreme line that still touches the feasible region. As we slide this line across the feasible region to find the maximum or minimum value, the last point of contact (or the first point of contact) with the boundary of the feasible region will always include at least one vertex point.

4. Conclusion: Therefore, to find the maximum or minimum value of the objective function, we evaluate the function at the vertex points (corner points) of the boundary of the feasible region. This makes "Vertex point of the boundary of the feasible region" the correct choice.