I'm some difficulty developing intuition for saddle-points. I would like to see an example of a smooth function $f:\mathbb{R}^2\rightarrow\mathbb{R}$ for which
$$\min_x\max_yf(x,y) > \max_y\min_xf(x,y)$$
A trivial example is $f(x,y) = \sin(x+y)$ (https://math.stackexchange.com/a/186746/10063). However in this case the min-max and max-min are not isolated points so first-order and second-order locality conditions do not characterize the saddle-point completely.
So I would like to see an example where the min-max and max-min are different isolated points, contained inside the domain (not on the boundary), so that first and second-order optimality conditions are meaningful.