I have to select one option from the problem statement given below.
Which of the following statements is true in case of linear programming.
$1$: An optimal solution exists at extreme points of a set of feasible solutions.
$2$: An optimal solution gives a hyperplane which is a supporting hyperplane to the set of feasible solutions.
$3$. Extreme points and basic feasible solutions are in one-one correspondence.
$4$: A set of feasible solutions is not necessarily a convex set.
My answer: Option $1$ is correct as optimal solution exists at extreme points of a set of feasible solutions.
Option $3$ is wrong as set of feasible solutions may be infinite while extreme points are finite. Option $4$ is wrong as set of feasible solutions are convex set. I am not sure about the option $2$ whether. However, answer key says that only option $2$ is correct. Could somebody explain me which option would be correct? Thank you very much.