I have a generated 3D surface, $z_1 = f^1(w(x),x,y)$ to look at the approximate magnitude of change for different combinations of $x_1$ and $y_1$. Here, $w(x)$ is the series of single positive roots of a cubic function $v(x)$ of $x$ where $a$, $b$ and $c$ are calculated according the initial conditions. $$v(x) = x^3 +a.x^2 +b.x + c$$ I then look at the same surface given a different set of starting conditions $z^2 = f^2(w(x),x,y)$ and take the difference of these two surfaces to end up with an idea of the trend observed upon changing initial conditions.
I noticed an unexpected saddle point with no obvious physical explanation and would like to show that this local minimum (if we take a slice at constant $y$) is expected behavior mathematically.
A surface slice at constant $y$ is given by $$z(w(x),x) = \frac{C.w(x)^2}{(w(x)+x)^2}$$ where $C$ is a constant. The difference of two at different initial conditions gives $$f(z_1-z_2) = \frac{C.w_2(x)^2}{(w_2(x)+x)^2} - \frac{C.w_1(x)^2}{(w_1(x)+x)^2}$$
This seems like an overly complicated expression for a 2D slice of my surface. My own research suggested that for multivariable functions can best be described by Jacobian or Hessian matrices. This seemed quite challenging given my expressions so I opted to take a 2D slice and at constant $y$ an solve for the derivative to obtain a local minimum, yet the equations I end up with are still very complicated.
My questions are the following: Is there any way to simplify the way I am considering the problem? (could I replace w(x) with a vector matrix of the precalculated roots for example?)
Are there simpler methods to analyze surface plots and saddle points that could help me demonstrate that the governing equations should be expected to show this saddle point with the specific combination of variables?
Are there known solutions to these types of problems? (I was not able to find any)