If U(X,Y,Z) is continuous, smooth, monotonic, and its first and second derivatives are monotonic, I can’t imagine how the linear approximation could fail.
There’s an example later in the post, with mixed derivatives. Everything could be smooth and monotonic including all derivatives. Basically think of U(X,Y,Z) as containing a 100XY component.
There’s an example later in the post, with mixed derivatives. Everything could be smooth and monotonic including all derivatives. Basically think of U(X,Y,Z) as containing a 100XY component.