Question: Extract boundary points from the inequalities

I am trying to find local extrema for multi variable functions. Using Hessian matrix and eigen values I am able to find the global extrema. Also by using boundary conditions I am able to solve for critical points with in given domain.

But, when there is no maxima or minima inside a local domain, It is believed to be minima/maxima lies on one of the boundaries(that point cannot be a critical point). I want to add this boundary points to the list of critical points

Example:

Term := x^3+x^2*y-2*y^3+6*y;
critical points := [[x = .6928203232, y = -1.039230485], [x = -.6928203232, y = 1.039230485], [x = 0., y = -1.], [x = 0., y = 1.]]
boundaries := [[-1<=x],[ x<=1], [-1<=y], [y<=1]];

now I want to read the boundaries as input and get the output as
boundaries :=[[x = -1,y =0],[x = 1,y =0],[x = 0,y =-1],[x = 0,y =1]];

When I did this manually I observed boundary points are saddle, since eigen values are mixed positive and negitive. But, my interest is to find the function value at boundaries.

Note: I believing value of other variables at perticular boundary is zero. In general I have to deal with multivariable functions with more than 3 variable.


Thank you