What is the graphical interpretation of the mixed partials? The partial derivative is pretty straightforward graphically, but I don't know about the mixed. Is it just a measure of the magnitude of the slope of the plane tangent to the surface at a given point?

Is there even a graphical significance? The only thing I've seen thusfar is that it is one of the quantities looked at when deciding if a point is a max or min.

The mixed partials of the D-Test are a cause of the Second Derivative Test Discriminant. They kind of explain its Geometrical Interpretation in the wiki but I'm still trying to decipher the underlying impact! I have never found a thorough derivation of the D-Test and it is something I kind of want to see...

The second order derivatives of a continuous function is known as the Hessian matrix of that function.

 

These are important in Morse theory, classical mechanics and the Legendre transformation, symplectic geometry and other things. I will have to have a think about any nice geometric picture of what is going on.